text:
\section{Introduction}

The Navier-Stokes equations are a system of partial differential equations that describe the evolution of a viscous incompressible fluid's velocity field $v$ and associated pressure $\pi$.  In three dimensional space they are
\begin{equation}
\begin{array}{ll}\label{eq:NSE}
 \partial_t v -\Delta v +v\cdot\nabla v+\nabla \pi  = 0
\\[5pt]   \nabla\cdot v = 0
\end{array}\quad \mbox{~in~}{\mathbb R }^3\times [0,\infty),
\end{equation}
and are supplemented with some initial data $v_0$. If the nonlinearity $v\cdot\nabla v$ is omitted, this becomes Stokes system.

Leray proved in \cite{leray} that, if $v_0\in L^2$, then a global in time weak solution $v$ to \eqref{eq:NSE} exists.  Hopf later generalized this to bounded domains (where the problem is supplemented with an appropriate boundary condition) in \cite{hopf}. Leray's construction is based on the a priori bound
\begin{equation}\label{ineq.energy}
\esssup_{0<t'<t}\int |v(x,t')|^2dx + \int_0^t \int 2 |\nabla v(x,t')|^2 dx\,dt'\le \int |v_0(x)|^2dx.
\end{equation}
Formally this is a result of testing \eqref{eq:NSE} against a solution $v$ and noting that the nonlinear term vanishes due to incompressibility.
This energy inequality is identical to that satisfied by solutions to the Stokes system.  Neither Leray nor Hopf were able to say much more  about these weak solutions; this remains true of researchers today.  In particular, we still do not know if Leray's weak solutions are unique or if they are smooth, even with smooth, compactly supported  $v_0$ (partial and conditional results are available, but the general questions remain open).

Leray noticed that solutions to \eqref{eq:NSE} satisfy a special scaling property: given a solution $v$ of \eqref{eq:NSE}, and $\lambda>0$, it follows that
\begin{equation}
	v^{\lambda}(x,t)=\lambda v(\lambda x,\lambda^2t),
\end{equation}
is also a solution with associated pressure
\begin{equation}
	\pi^{\lambda}(x,t)=\lambda^2 \pi(\lambda x,\lambda^2t),
\end{equation}
and initial data
\begin{equation}
v_0^{\lambda}(x)=\lambda v_0(\lambda x).
\end{equation}
When weak solutions to a system are not clearly regular, irregular behavior can sometimes be found by considering special solutions. Along these lines Leray realized that solutions which are invariant to the above scaling, if defined for negative times, would necessarily develop a singularity at time $t=0$.

We say a solution is self-similar (SS) if it is scaling invariant with respect to the above scaling, i.e.~if $v^\lambda(x,t)=v(x,t)$ for all $\lambda>0$. Hence, Leray's proposed singular solutions would be self-similar.
If this scale invariance holds for a particular $\lambda>1$, then we say $v$ is discretely self-similar with factor $\lambda$ (i.e.~$v$ is $\lambda$-DSS). Similarly $v_0$ can be SS or $\lambda$-DSS.  The class of DSS solutions contains the SS solutions since any SS $v$ is $\lambda$-DSS for any $\lambda>1$.

Self-similar solutions can be forward (i.e.~defined for $t>0$) or backward (i.e.~defined for $t<0$, like Leray's proposed solutions), and both classes are interesting as sources of irregular behavior such as singularity formation and non-uniqueness.

For the backward case,
as mentioned above, Leray \cite{leray} proposed the SS solutions as candidates for singularity formation. His original problem on the existence of a nontrivial solution $v(\cdot,t)$ in $W^{1,2}({\mathbb R }^3) \subset L^3({\mathbb R }^3)$ for every $t<0$ was excluded in Ne\v cas, {R\accent23 u\v {z}i\v {c}ka}, and {\v Sver\'ak} in \cite{NRS}.   Escauriaza, Seregin, and {\v Sver\'ak} \cite{ESS} gave another proof of the result in \cite{NRS} as a consequence of their $L^\infty L^3$ regularity criteria. Tsai proved a localized non-existence result in \cite{Tsai-ARMA} for solutions $v$ satisfying $v(t) \in L^q({\mathbb R }^3)$, $3<q\le \infty$ for any $t<0$, or $v \in L^{10/3}(B_1 \times (-1,0))$.  These results were recently generalized to Lorentz spaces by Chae and Wolf in \cite{Chae-Wolf2} and by Guevara and Phuc in \cite{GuePhuc}.  Chae and Wolf also proved the non-existence of non-trivial backward \emph{discretely} self-similar solutions \cite{Chae-Wolf3} where the scaling factor $\lambda$ is close to $1$, (with $N=\sup_{x,t}(x^2-t)^{1/2}|v(x,t)|<\infty$ and
$\lambda-1\le \delta$ for some sufficiently small $\delta>0$ depending on $N$), by reducing the problem to the self-similar case via a contradiction argument.
There are still several important open problems concerning the existence of backward solutions in larger classes.  For example, solutions that are self-similar modulo a rotation (these will be introduced formally later in this article) do not exist if $u \in L^\infty(-1,0; L^3({\mathbb R }^3))$ (this follows from the result of \cite{ESS}), but this is not known under the weaker assumption
\[
|u(x,t)| \le \frac C{|x|+\sqrt{-t}} \quad \text{in }{\mathbb R }^3 \times (-1,0),
\]
even though this assumption does exclude backward self-similar solutions \cite{Tsai-ARMA}.  This is surprising because, at face value, rotated self-similar solutions appear very similar to self-similar solutions (e.g.~they both have stationary ansatzes).  Additionally, backward DSS solutions haven't been ruled out under any condition when $\lambda$ is significantly larger than one.

For the forward case,
it has been conjectured that {forward} self-similar solutions are a good place to look for non-uniqueness of Leray weak solutions to the Navier-Stokes equations \cite{JiaSverak,JiaSverak2}.
Self-similar solutions are decided by their behavior at a single time and can therefore be determined by an ansatz which satisfies a stationary system resembling the stationary Navier-Stokes equations.   It is known for certain large data and appropriate forcing, that solutions to the stationary Navier-Stokes boundary value problem are non-unique \cite{Velte,Yudovich,Galdi,Temam}.  In \cite{JiaSverak}, Jia and \v Sver\'ak speculate that similar non-uniqueness results might hold for the stationary profiles of \emph{forward} self-similar solutions, provided the initial data is large in a borderline space such as $L^3_w$. Strong numerical evidence has recently been given by Guillod and \v Sver\'ak  in \cite{GuiSve} supporting the non-uniqueness conjecture in these spaces.

Until recently, existence results for forward self-similar and DSS solutions were only known for small data (for small data existence of forward self-similar solutions see \cite{GiMi,Kato,Barraza,CP,Koch-Tataru}).  Such solutions are unique and, therefore, the scale invariance of the solution follows from that of the initial data.
For large data there is no uniqueness result.
 Thus, an existence theory of scale invariant solutions for large data is needed. This article will survey recent advances toward a robust existence theory for forward self-similar and discretely self-similar solutions with large data.  As we will see, the problem is largely solved, but several important outstanding cases remain open.

This article is structured as follows.  In Section \ref{sec.properties} we will introduce and discuss several classes of solutions to the Navier-Stokes equations with special symmetries (self-similar solutions being an example of such a class).  In Section \ref{sec.classes} we introduce classes of solutions to the Navier-Stokes equations, including ones for initial data that is uniformly locally square integrable or belongs to scaling invariant function spaces.
In Section \ref{sec.overview} we will give an overview of the various constructions of solutions with special symmetries that are presently available.  In Section \ref{sec.construction} we will focus on the construction from \cite{BT1,BT2,BT3} and use it to prove Theorem \ref{thrm.new}, which is an improvement of the existence result in \cite{BT3}.  In particular, Theorem \ref{thrm.new} establishes a local energy inequality for the SS/DSS solution constructed by the method from \cite{BT1,BT3} for initial data in Besov spaces of negative order.

\section{Properties of scaling invariant solutions}\label{sec.properties}

Recall that $v$ is self-similar (SS) if $v(x,t)=\lambda v(\lambda x,\lambda^2 t)$ for all $\lambda>0$ and $v_0$ is SS if $v_0(x)=\lambda v_0(\lambda x)$ for all $\lambda>0$.
If this holds for a particular $\lambda>1$, then $v$ (or $v_0$) is $\lambda$-DSS.  Clearly, if $v$ is SS then it is $\lambda$-DSS for all $\lambda>1$. Of course, a solution can also be strictly discretely self-similar.

Self-similar solutions have a stationary quality in that there exists an ansatz for $v$ in terms of a time-independent profile $u$, namely,
\begin{equation}\label{ansatz1}
v(x,t) = \frac 1 {\sqrt {t}}\,u\bigg(\frac x {\sqrt{t}}\bigg).
\end{equation}
The above applies to forward solutions, and we replace $t$ by $-t$ for backward solutions.
The profile $u$ solves the \emph{Leray equations}
\begin{equation}
\begin{array}{ll}\label{eq:stationaryLeray}
 -\Delta u-\frac 1 2 u-\frac 1 2 y\cdot\nabla u +u\cdot \nabla u +\nabla p = 0&%
\\[5pt]  \nabla\cdot u=0&%
\end{array}
\mbox{~in~}{\mathbb R }^3,
\end{equation}
in the variable $y=x/\sqrt{ t}$.    Note that these equations closely resemble the stationary Navier-Stokes equations.

Discretely self-similar solutions on the other hand are determined by their behavior on time intervals of the form $\lambda^k\leq t\leq \lambda^{2+k}$.  Thus they have a periodic quality. To make this more precise consider the ansatz
\begin{equation}\label{ansatz2}
v(x,t)=\frac 1 {\sqrt{t}}\, u(y,s),\quad  \pi(x,t) = \frac 1{ t} p(y,s),
\end{equation}
where
\begin{equation}\label{variables2}
y=\frac x {\sqrt{t}},\quad s=\log t.
\end{equation}
Then, the vector field $u$ is $T$-periodic in $s$ with period $T=2\log \lambda$ and solves the \emph{time-dependent Leray equations}
\begin{equation}
\begin{array}{ll}
\label{eq:timeDependentLeray}
 \partial_s u-\Delta u-\frac 1 2 u-\frac 1 2 y\cdot\nabla u +u\cdot \nabla u +\nabla p = 0&
\\[5pt]   \nabla\cdot u = 0&
\end{array}
\mbox{~in~}{\mathbb R }^3\times {\mathbb R },
\end{equation}
where ${\nabla} = {\nabla}_y$ and $\Delta = \Delta_y$.
Note that the \emph{similarity transform} \eqref{ansatz2}--\eqref{variables2} gives a one-to-one correspondence between solutions to \eqref{eq:NSE} and \eqref{eq:timeDependentLeray}.

A rotational correction can be added to self-similar and discretely self-similar symmetries. For ease of notation,
we will only consider rotations around the $x_3$-axis with matrices
\[
R_s=R( s) = \mat{
\cos s & -\sin s & 0 \\
\sin s & \cos s & 0 \\
0 & 0 & 1 }.\]
Note $R( s)R(\tau)=R(\tau)R( s)$ for any $s,\tau \in {\mathbb R }$, and
\[
\frac d{ds} R(s) = JR(s) = R(s)J, \quad
J= \mat{
0 & -1 & 0 \\
1&0 & 0 \\
0 & 0 & 0 }.\]

A vector field $v(x,t)$ is said to be \emph{rotated self-similar} (RSS) if, for some fixed $\alpha\in {\mathbb R }$ and for \emph{all} $\lambda>0$, $x\in {\mathbb R }^3$, and $t>0$,
\begin{equation}
\label{v-RSS}
v(x,t) = \lambda R(- 2\alpha \log \lambda)\, v\! \bke{\lambda R(2 \alpha \log \lambda) x, \lambda^2 t}.
\end{equation}
The constant $\alpha$ will be called the \emph{angular speed}, and is understood relative to the new time variable $s$ to be defined in \eqref{eq.v.to.u}.
An RSS vector field is always DSS with any factor $\lambda>1 $ such that $2 \alpha \log \lambda \in 2\pi \mathbb{Z}$.
When $\alpha=0$ it becomes SS.  Hence SS $\subsetneq$ RSS $\subsetneq$ DSS.

Setting $\lambda=t^{-1/2}$, any RSS vector field $v$ satisfies
\EQ{
v(x,t) = R(\alpha \log {t}) \frac 1{\sqrt{t}}\, v\!\bke{R(-\alpha \log {t}) \frac x{\sqrt{t}} , 1},
}
for all $x\in{\mathbb R }^3$ and $t>0$.
Thus the value of $v$ is determined by its value at any fixed time, and given any profile at a fixed time we can construct an RSS vector field.  Hence RSS solutions have the same stationary quality as SS solutions.

A vector field $v(x,t)$ is said to be \emph{rotated discretely self-similar} (RDSS) if, for some $\lambda>1$ (not necessarily all $\lambda>1$)
and some $\phi \in {\mathbb R }$,
\begin{equation}
\label{v-RDSS}
v(x,t) = \lambda R(-\phi)\, v\! \bke{\lambda R(\phi) x, \lambda^2 t},
\end{equation}
for all $x\in{\mathbb R }^3$ and $t>0$.
We call $\lambda$ the \emph{factor} and $\phi$ the \emph{phase}.
When $\phi\in 2\pi \mathbb{Z}$ we recover $\lambda$-DSS vector fields.
If $n\phi=2\pi m$ for some integers $n>0$ and $m$, then $v$ is DSS with factor $\lambda^n$. If $\frac\phi{2\pi}$ is irrational, in general $v$ is not DSS.
For any $t>0$ let $\tau(t)\in [1,\lambda^2)$ satisfy $\tau=\lambda^{2k}t$ for some $k\in {\mathbb Z}$.  Then,
\[
v(x,t)= \lambda^k R(-k\phi)\, v\! \bke{\lambda^k R(k \phi) x, \tau},
\]
i.e., $v$ is decided entirely by its values on $t\in [1,\lambda^2)$.
{Note that an RSS vector field with angular speed $\alpha$ is always RDSS for any factor $\lambda>1$ with phase $\phi=2\alpha \log \lambda $.}

In summary, the inclusions between these classes are
\[
\mbox{SS}\subsetneq \mbox{RSS} \subsetneq \mbox{DSS} \subsetneq \mbox{RDSS}.
\]

For initial data, we say a vector field
$v_0(x):{\mathbb R }^3\to {\mathbb R }^3$ is RSS if, for some $\alpha\in {\mathbb R }$ and all $x\in{\mathbb R }^3$ and $\lambda>0$,
\begin{equation}
\label{v0-RSS}
v_0(x) = \lambda R(- 2 \alpha \log \lambda) v_0\bke{\lambda R( 2\alpha \log \lambda) x},
\end{equation}
and is
RDSS if for some $\lambda>1$ and some $\phi \in {\mathbb R }$,
\begin{equation}
\label{v0-RDSS}
v_0(x) = \lambda R(-\phi) v_0\bke{\lambda R(\phi) x},
\end{equation}
for all $x\in {\mathbb R }^3$.
Note that, like SS initial data, if $v_0$ is RSS then it is determined by its values on the unit sphere. Similarly, if $v_0$ is RDSS, then it is determined by its values on $\{ x: 1\leq |x|<\lambda \}$.

RSS and RDSS solutions also have ansatzes which are stationary or time-periodic.  Indeed, let
\begin{equation}
\label{eq.v.to.u}
v(x,t) =\frac 1{\sqrt t} R_\theta u(y,s), \quad  \pi(x,t) = \frac 1{ t} p(y,s),\quad y = R_\theta^T \frac x {\sqrt t},\quad s=\log t,
\end{equation}
for some function $\th(s)$. Then $u$ satisfies
\EQ{\label{eq:Leray-rot}
  \partial_s u  + \dot \th J u -  (\dot \th J y)\cdot {\nabla} u
  &-  \frac 12 u -  \frac y2\cdot\nabla u -\Delta u +u\cdot\nabla u+\nabla p  = 0,
 \\ & \nabla \cdot u = 0,
}
in ${\mathbb R }^3\times {\mathbb R }$.
If $v(x,t)$ is an RSS solution of \eqref{eq:NSE} satisfying \eqref{v-RSS}, then $u(y,s)$ is a stationary solution of \eqref{eq:Leray-rot} with constant $\dot \theta = \alpha$, if we take  $\theta(s)=\alpha s$.
For any RDSS solution $v(x,t)$ of \eqref{eq:NSE} satisfying \eqref{v-RDSS} with factor $\lambda>1$ and phase $\phi$,
let
\begin{equation}
\label{alpha-choice}
T=2 \log \lambda,\quad
 \alpha_k= \frac {2k\pi + \phi}T ,
\end{equation}
for an arbitrary integer $k\in \mathbb{Z}$. Then $v(x,t)$
corresponds to a periodic solution $u(y,s)$ of \eqref{eq:Leray-rot} with constant $\dot \theta = \alpha_k$ and period $T$. To be definite we will take $\alpha=\alpha_0 = \frac \phi T$.

\section{Solution classes}\label{sec.classes}

In this section we introduce various function spaces and classes of solutions to the Navier-Stokes equations.  These will be highly relevant to our subsequent discussion of self-similar solutions and their generalizations.

\subsection{Weak solutions}

As mentioned in the introduction, Leray constructed weak solutions to the Navier-Stokes equations in \cite{leray}.  His solutions satisfied the global energy inequality \eqref{ineq.energy}.  In honor of Leray's contribution, any weak solution with data in $L^2$ satisfying \eqref{ineq.energy} is called a \emph{Leray weak solution}.

Lemari\'e-Rieusset introduced a local analogue of Leray's solutions in \cite{LR} called \emph{local Leray solutions}.  We recall the definition in full.  For $q \in [1,\infty)$, we say $f \in L^q_\mathrm{uloc}$ if $f$ is defined in ${\mathbb R }^3$ and
\[
\norm{f}_{L^q_\mathrm{uloc}} = \sup_{x_0\in {\mathbb R }^3} \norm{f}_{L^q(B_1(x_0))}<\infty.
\]

\begin{definition}[Local Leray solutions]\label{def:localLeray} A vector field $v\in L^2_{\loc}({\mathbb R }^3\times [0,\infty))$ is a local Leray solution to \eqref{eq:NSE} with divergence free initial data $v_0\in L^2_{\mathrm{uloc}}$ (and zero force) if:
\begin{enumerate}[label=$($\alph*$)$]
\item for some $\pi\in L^{3/2}_{\loc}({\mathbb R }^3\times [0,\infty))$, the pair $(v,\pi)$ is a distributional solution to \eqref{eq:NSE},
\item for any $R>0$, $v$ satisfies
\begin{equation}\notag
\esssup_{0\leq t<R^2}\,\sup_{x_0\in {\mathbb R }^3}\, \int_{B_R(x_0 )} |v(x,t)|^2\,dx + \sup_{x_0\in {\mathbb R }^3}\int_0^{R^2}\! \!\int_{B_R(x_0)} |\nabla v(x,t)|^2\,dx \,dt<\infty,\end{equation}
\item for any $R>0$, $v$ satisfies
\begin{equation}\notag
\lim_{|x_0|\to \infty} \int_0^{R^2}\!\! \int_{B_R(x_0 )} | v(x,t)|^2\,dx \,dt=0,
\end{equation}
\item for all compact subsets $K$ of ${\mathbb R }^3$ we have $v(t)\to v_0$ in $L^2(K)$ as $t\to 0^+$,
\item $v$ is suitable in the sense of Caffarelli-Kohn-Nirenberg, i.e., for all cylinders $Q$ compactly supported in  $ {\mathbb R }^3\times(0,\infty )$ and all non-negative $\phi\in C_0^\infty (Q)$, we have
\EQ{\label{ineq.localv}
&%
2\iint |\nabla v|^2\phi\,dx\,dt
\\&\leq
\iint |v|^2(\partial_t \phi + \Delta\phi )\,dx\,dt +\iint (|v|^2+2\pi)(v\cdot \nabla\phi)\,dx\,dt.
}
\end{enumerate}
\end{definition}

In \cite{LR} Lemari\'e-Rieusset constructed global in time  local Leray solutions if $v_0$ belongs to $E_2$, the closure of $C_0^\infty$ in the $L^2_{\mathrm{uloc}}({\mathbb R }^3)$ norm.  See Kikuchi-Seregin \cite{KiSe} for more details. In particular, condition 3 justifies a formula of the pressure $\pi$ in terms of the velocity $v$, see \cite[(1.9)]{KiSe} and \cite[(3.3)]{JiaSverak}.

Local Leray solutions are known to satisfy a useful a priori bound.  Let $\mathcal N (v_0)$ denote the set of all local Leray solutions with initial data $v_0$. The following estimate is known for local Leray solutions (see \cite{JiaSverak}): for all $\tilde v\in \mathcal N (v_0)$ and $r>0$ we have
\begin{equation}\label{ineq.apriorilocal}
\esssup_{0\leq t \leq \sigma r^2}\sup_{x_0\in {\mathbb R }^3} \int_{B_r(x_0)}\frac {|\tilde v|^2} 2 \,dx\,dt + \sup_{x_0\in {\mathbb R }^3}	\int_0^{\sigma r^2}\!\! \int_{B_r(x_0)} |\nabla \tilde v|^2\,dx\,dt <C A _r,
\end{equation}
where
\begin{equation}
\label{ineq.apriorilocalconstant}
A_r= \sup_{x_0\in {\mathbb R }^3} \int_{B_r(x_0)} \frac {|v_0|^2}2\,dx, \quad
\sigma(r) =c_0\, \min\big\{r^2A_r^{-2} , 1  \big\},
\end{equation}
for a small universal constant $c_0$.

Local Leray solutions are currently only defined on the whole space because the pressure formula breaks down on domains possessing boundaries.

\subsection{Mild solutions}

There is a rich literature about the global well-posedness of solutions to the Navier-Stokes equations when the data belongs to a scaling invariant function space (see \cite{LR} for a review of this).  Note that $X$ is scaling invariant if $\|u\|_X= \|u^{\lambda} \|_X$ for any $\lambda$.  Since we are working over ${\mathbb R }^3$, the most obvious example of a scaling invariant function space is $L^3$.  Global well-posedness is known in $L^3$ for small data. But, $L^3$ does not include non-trivial self-similar data because any such data will look like $|x|^{-1}$ on rays emanating from the origin.

The weak Lebesgue space $L^3_w$ is slightly larger than $L^3$ and includes $|x|^{-1}$, making it a natural space to study self-similar initial data.  Recall that $f\in L^3_w({\mathbb R }^3)$ if and only if
$\|f\|_{L^3_w({\mathbb R }^3)}<\infty$,
where
\begin{equation}\label{weakL3}
\|f\|_{L^3_w({\mathbb R }^3)} = \sup_{s>0}\,s \,m(f,s)^{1/3},
\end{equation}
and $m(f,s)$ is the distribution function of $f$ given by
\[
m(f,s)=|\{ x\in {\mathbb R }^3: | 	f(x)|>s  \}|.
\]
Here $|S|$ denotes the three dimensional Lebesgue measure of a given set $S$.
Let $L^3_{w,\sigma}({\mathbb R }^3)$ be the subspace of $L^3_w$ of divergence free vector fields.
In $L^3_w$ the best global well-posedness result is the following: if $u_0\in L^3_{w,\sigma}$ and $\|u_0\|_{L^3_w}<\epsilon_0$ for a universal, small constant $\epsilon_0$, then there exists a global in time \emph{mild} solution $u$ which is also a strong (i.e.~classical) solution and is unique among solutions belonging to $L^\infty(0,\infty;L^{3}_w)$ \cite{Barraza,LR}.  By mild solution we mean a solution to  \eqref{eq:NSE} satisfying an integral formulation.  Such solutions are nonlinear perturbations of $e^{t\Delta}v_0$.

Note that $L^3_w({\mathbb R }^3)$ embeds continuously into the space of uniformly locally square integrable functions $L^2_{\mathrm{uloc}}({\mathbb R }^3)$.  Thus, for any initial data in $L^3_{w,\sigma}$, one may construct a global-in-time {local Leray solutions} in the whole space. However, this solution
may not be unique for large data, and hence is not necessarily self-similar.

Analogous wellposedness results for small data can be formulated in the Besov spaces $\dot B_{p,\infty}^{3/p-1}$ ($1<p<\infty$) and the Koch-Tataru space $BMO^{-1}$, which we now introduce.  Besov spaces can be defined using the Littlewood-Paley decomposition.  Fix an inverse length scale $\lambda>1$.
Let $B_r$ denote the ball of radius $r$ centered at the origin in ${\mathbb R }^3$.  Fix a non-negative, radial cut-off function $\chi\in C_0^\infty(B_{1})$ so that $\chi(\xi)=1$ for all $\xi\in B_{1/\lambda}$. Let $\phi(\xi)=\chi(\lambda^{-1}\xi)-\chi(\xi)$ and $\phi_j(\xi)=\phi(\lambda^{-j}\xi)$.  For a vector field $u$ of tempered distribution, let $\Delta_j u=(\mathcal F^{-1}\phi_j)*u$ for $j\in {\mathbb N}_0 $ and $\Delta_{-1}=(\mathcal F^{-1}\chi)*u$. Then, $u$ can be written as\[u=\sum_{j\geq -1}\Delta_j u.\]
If $(\mathcal F^{-1}\chi(\lambda^{-j}\cdot))*u\to 0$ as $j\to -\infty$ in the space of tempered distributions, then for $j\in {\mathbb Z}$ we define $\dot \Delta_j u = \mathcal F^{-1}\phi_j*u$ and have
\[u=\sum_{j\in {\mathbb Z}}\dot \Delta_j u.\]
For $s\in {\mathbb R }$, $1\leq p,q\leq \infty$, the non-homogeneous Besov spaces include tempered distributions modulo polynomials for which the norm
\begin{align*}
&\|u\|_{B^s_{p,q}}:=
\begin{cases}
 \bigg(\sum_{ j\geq -1} \big(    \lambda^{sj} \|\Delta_j u \|_{L^p({\mathbb R }^n)}  \big)^q \bigg)^{1/q}   & \text{ if } q<\infty
\\ \sup_{j\geq -1} \lambda^{sj} \| \Delta_j u \|_{L^p({\mathbb R }^n)} & \text{ if } q=\infty
\end{cases},
\end{align*}is finite, while the homogeneous Besov spaces include tempered distributions modulo polynomials for which the norm
\begin{align*}
&\|u\|_{\dot B^s_{p,q}}:=
\begin{cases}
 \bigg(\sum_{ j\in {\mathbb Z}} \big(    \lambda^{sj} \|\dot \Delta_j u \|_{L^p({\mathbb R }^n)}  \big)^q \bigg)^{1/q}   & \text{ if } q<\infty
\\ \sup_{j\in {\mathbb Z}} \lambda^{sj} \| \dot \Delta_j u \|_{L^p({\mathbb R }^n)} & \text{ if } q=\infty
\end{cases},
\end{align*}
is finite.
Note that the partition of unity can be taken to be $\lambda$-adic for any $\lambda>1$.  The resulting Besov norm is equivalent to any dyadic norm \cite{BT3}.
For $3\leq p<\infty$, a small data global well-posedness theory exists for data in $\dot B_{p,\infty}^{3/p-1}$ that is identical to that for data in $L^3_w$ \cite{BCD} (with $L^3_w$ replaced by $\dot B_{p,\infty}^{3/p-1}$).  Since $L^3_w\subsetneq \dot B_{p,\infty}^{3/p-1}$ whenever $p>3$, this generalizes the result of \cite{Barraza}.

The Koch-Tataru space $BMO^{-1}$ is the largest critical function space in which the Navier-Stokes equations are globally well-posed for small data and consists of distributions which are derivatives of elements of $BMO$, the class of functions with bounded mean oscillation.  The existence of global in time mild strong solutions was proved by Koch and Tataru in \cite{Koch-Tataru}.
Note that $\dot B_{p,\infty}^{3/p-1}\subsetneq BMO^{-1} \subsetneq \dot B_{\infty,\infty}^{-1}$.  We have observed that small data well-posedness holds for the first two spaces; it fails for the third:  In \cite{BoPa}, Bourgain and Pavlovic proved the last point by showing that, for any time length $\delta$, any $\epsilon>0$, and any $M>0$, there exists an initial data $u_0$ so that $\|u_0\|_{\dot B_{\infty,\infty}^{-1}}<\epsilon$ and $\|u(\delta)\|_{\dot B_{\infty,\infty}^{-1}}>M$ (this is called norm inflation).

\section{Existence results} \label{sec.overview}

\subsection{Strong self-similar solutions for small data}

In the previous section we mentioned global-in-time well posedness is known for small data in certain critical spaces including $L^3_w$, $\dot B_{p,\infty}^{3/p-1}$, and $BMO^{-1}$.  These results give the existence of SS/RSS/DSS/RDSS solutions as a corollary.  Indeed, assume that $v_0$ is SS and $\|v_0\|_X$ is small where $X$ is any of the spaces just mentioned.  Then, there exists a strong mild solution $v$ to \eqref{eq:NSE} that belongs to and is unique in $C_w([0,\infty);X)$.
  But, for every $\lambda>0$, $v^{\lambda }$ is also a solution and belongs to $C_w([0,\infty);X)$. Since $v_0$ is SS, $v_0^{\lambda }=v_0$ and, by uniqueness in $C_w([0,\infty);X)$, it follows that $v=v^{\lambda}$ for all $\lambda$, i.e.~$v$ is self-similar.  The same argument goes through for RSS, DSS, and RDSS data.

For small data, self-similar solutions were studied in the eighties and nineties by \cite{GiMi, Kato, Barraza, CP, Koch-Tataru}.
Giga-Miyakawa \cite{GiMi} constructed solutions to the vorticity equations assuming the initial vorticity is in a critical space of measures.
Kato gave the construction in \cite{Kato} assuming the initial velocity was small in a critical Morrey space.
 In 1996, Barraza \cite{Barraza} constructed solutions for small data in $L^3_w$, as did Cannone and Planchon \cite{CP} for small data in $\dot B_{p,\infty}^{3/p-1}$ where $p>3$.  Finally the solution is constructed in $BMO^{-1}$ by Koch-Tataru \cite{Koch-Tataru}.
The regularity properties of Koch-Tataru  solutions are studied by Miura and Sawada \cite{MiuraSawada} and by
 Germain, Pavlovi\'c, and Staffilani \cite{GePaSt}.
Decay and asymptotic properties of small self-similar solutions have been further examined by Brandolese in \cite{Bran}.

\subsection{Strong self-similar solutions for large, smooth data}

As mentioned above, global well-posedness is only known in $L^3_w$ for small data.  For large data, only weak solutions are known to exist.  Indeed, since $L^3_w$ is in the closure of $C_0^\infty$ under $L^2_{\mathrm{uloc}}$-norm, for any $v_0\in L^3_w$, there exists a global in time local Leray solution $v$ \cite{KiSe,LR}.  Barker and Seregin have recently given another approach to constructing global solutions for large data in $L^3_w$ \cite{BaSe}.  Because uniqueness is not known in these classes, it is unclear if the solutions constructed in \cite{KiSe,LR,BaSe} inherit the scaling properties of the initial data.  Thus, for large data in $L^3_w$, we do not get the existence of self-similar solutions for free by these methods.

It was thus surprising when, in 2014, Jia and \v Sver\'ak  constructed a forward self-similar solution for \emph{large $-1$-homogeneous} initial data which is \emph{locally H\"older continuous} away from the origin \cite{JiaSverak}.  Their proof used Leray-Schauder degree theory, the main ingredient of which are \emph{a priori} bounds for self-similar solutions, existence and uniqueness for small data, and global compactness.  Most of the work lies in establishing H\"older estimates for the solutions, which depend on the fact that the solutions live in the local Leray class.

In \cite{Tsai-DSSI}, Tsai gave a similar result for $\lambda$-DSS solutions with factor \emph{close to one} where closeness is determined by the local H\"older norm of $v_0$ away from the origin.  It is also shown in \cite{Tsai-DSSI} that the closeness condition on $\lambda$ can be eliminated if the initial data is axisymmetric with no swirl. The approach is similar to \cite{JiaSverak}.
In \cite{KT-SSHS}, Korobkov and Tsai constructed self-similar solutions on the half space (their approach also works on the whole space) for appropriately smooth initial data.  The approach here differs from \cite{JiaSverak} and \cite{Tsai-DSSI} in that the existence of a solution to the stationary Leray equations \eqref{eq:stationaryLeray} is established directly.  It also gives a second proof of the main result of \cite{JiaSverak}.  {A new approach is necessary in
\cite{KT-SSHS} due to lack of spatial decay estimates, which gives  \emph{global compactness} needed for the Leray-Schauder theorem in
\cite{JiaSverak} and \cite{Tsai-DSSI}.}

\subsection{Weak self-similar solutions for large, rough data}

The constructions in \cite{JiaSverak,Tsai-DSSI,KT-SSHS} necessarily produce strong solutions; this is why they require the initial data to be H\"older continuous.  If we instead  use construction methods which only necessarily yield weak solutions, then we can weaken the assumptions imposed on the initial data.  Indeed, this idea is the main motivation for the sequence of papers by the authors, \cite{BT1}, \cite{BT2}, and \cite{BT3}, where SS/RSS/DSS/RDSS solutions are constructed for large, possibly rough data.  In \cite{BT1} and \cite{BT2} we considered data in $L^3_w$ and arrived at the following theorem (which is an amalgam of the results of \cite{BT1,BT2}).

\begin{theorem}\label{thrm:BT1}
Let $v_0$ be a SS vector field and belong to $L^3_{w,\sigma}({\mathbb R }^3)$. Then, there exists a local Leray solution $v$ to \eqref{eq:NSE} which is SS and additionally satisfies
\begin{equation}\notag
 \|  v(t)-e^{t\Delta}v_0 \|_{L^2({\mathbb R }^3)}\leq C_0\,t^{1/4}
\end{equation}
for any $t\in (0,\infty)$ and some constant $C_0=C_0(v_0)$.

The above statement is also true with ``SS'' replaced by any of the following:
\begin{itemize}
\item ``RSS for any given $\alpha\in {\mathbb R }$'',
\item  ``DSS for any given $\lambda>1$'',
\item ``RDSS for any given $\alpha\in {\mathbb R }$ and $\lambda>1$''.
\end{itemize}
\end{theorem}

This result is more general than all earlier results.  In particular, the initial data can be discontinuous or even singular away from the origin.  Furthermore, it is valid for any choice of available parameters.  The price paid for this generality is a loss of \emph{a priori} regularity of the solutions; the solutions from \cite{BT1} are not guaranteed to be smooth by their construction.  In the SS and RSS cases, smoothness follows after the fact using Gruji\'c's result which implies that any SS local Leray solution is smooth \cite{Grujic} (that proof also works in the RSS case).  In the DSS and RDSS cases, smoothness is not known generally, but is expected for values of $\lambda$ close to $1$.

An analogous result can be formulated on the half space, as is shown in \cite{BT2}.  Adapting the proof of \cite{BT1} to that case requires several technical modifications.
In particular, the solution class is weakened, and we do not impose the local energy inequality.
 In comparison, self-similar solutions for smooth data on ${\mathbb R }^3_+$ were constructed in \cite{KT-SSHS} but strictly discretely self-similar solutions had not been constructed prior to \cite{BT2}.

Because existence of SS solutions is known for small data in spaces larger than $L^3_w$, in particular the Besov spaces $\dot B_{p,\infty}^{3/p-1}$, it is natural to try to extend Theorem \ref{thrm:BT1} in that direction.   Partial progress was made in \cite{BT3} to the Besov spaces $\dot B_{p,\infty}^{3/p-1}$ for $3<p<6$ for SS and DSS data.
The following theorem is a slight refinement of the main result of \cite{BT3}.

\begin{theorem}\label{thrm.new} Fix $p\in (3,6)$.  Assume $v_0:{\mathbb R }^3\to {\mathbb R }^3$ is divergence free, belongs to $\dot B_{p,\infty}^{3/p-1}$, and is self-similar. Then there exists a self-similar distributional solution $v$ and pressure distribution $\pi$ to 3D NSE on ${\mathbb R }^3\times (0,\infty)$.
Furthermore, $v$ and $v_0$ can be decomposed as $a+b$ and $a_0+b_0$ respectively so that $a_0\in L^3_w$, $b_0$ is small in $\dot B_{p,\infty}^{3/p-1}$, and $a$ and $b$ satisfy the following properties:
\begin{itemize}
\item $a$ and $b$ are self-similar,
\item $b$ is a global strong solution to \eqref{eq:NSE} for initial data $b_0$ having associated pressure $p_b$,
\item there exists a distribution $p_a$ so that $a$ and $p_a$ solve
\EQ{\label{eq.a}
\partial_t a- \Delta a + a\cdot\nabla a+ a\cdot\nabla b + b\cdot \nabla a +\nabla p_{a}=0,
\quad \div a=0,
}
in the sense of distributions,
\item $a(t)$ converges to $a_0$ in the sense that
\EQ{\label{th1.3-1}
\|a(t)-e^{t\Delta}a_0\|_{L^2} &\leq C_2 t^{1/4},\\
\int_{0}^t \norm{a(\tau)-e^{\tau\Delta}a_0}_{L^r}^q d\tau  &\le C_r t^{q/4},\quad \forall\, r \in (2,6],
}
for some constant $C_r(v_0)$ with $\frac 3r + \frac 2q=\frac 32$,
\item  $b(t)$ converges to $b_0$ in the sense that
\begin{equation}\label{th1.3-2}
\norm{b(t)-e^{t\Delta}b_0}_{L^r} \le C_r \norm{b_0}_{\dot B_{p,\infty}^{3/p-1}}^2 t^{-\frac 12 + \frac 3{2r}}, \quad \forall\, r \in
\left[\frac p2, \frac {3p}{6-p}\right),
\end{equation}
for some constant $C_r$,
\item  $v=a+b$ and $\pi = p_a + p_b$ satisfy the local energy inequality \eqref{ineq.localv}
for any
non-negative $\phi\in C^\infty_0({\mathbb R }^3\times {\mathbb R }^3_+)$.

\end{itemize}

The above statement is also true with ``self-similar'' replaced by {``DSS for any given $\lambda>1$''.}
\end{theorem}

We will prove Theorem \ref{thrm.new} in Section \ref{sec.construction}.

\medskip

 \noindent\emph{Comments on Theorem \ref{thrm.new}}:
\begin{enumerate}[label=(\alph*)]
\item Theorem \ref{thrm.new} improves  the main result of \cite{BT3} in that
it includes the local energy inequality \eqref{ineq.localv}.
 It is however restricted to the whole space setting, not for ${\mathbb R }^3_+$.
\item If $b_0$ is smooth, for $\phi \in C^\infty_0({\mathbb R }^3 \times [0,\infty))$, possibly non-zero at $t=0$,
we can replace $\phi$ in \eqref{ineq.localv}
by $\phi \theta(t/\epsilon)$, where $\theta(t)=1$ for $t>1$ and $\theta(t)=0$ for $t<1/2$, and take limits $\epsilon \to 0_+$ to get the usual form of local energy inequality %
\EQ{ \label{ineq.localv-2}
& 2\iint |\nabla v|^2\phi\,dx\,dt
\leq \int |v_0|^2 \phi|_{t=0} dx
\\
&\quad+ \iint |v|^2(\partial_t \phi + \Delta\phi )\,dx\,dt
 +\iint (|v|^2+2p)(v\cdot \nabla\phi)\,dx\,dt.
}
For general $b_0 \in \dot B_{p,\infty}^{3/p-1}$, however, $b_0$ may not be in $L^2_\loc$ (see \cite[\S6]{BT3}), hence \eqref{ineq.localv-2} is not meaningful.
\end{enumerate}

Theorem \ref{thrm:BT1} has also been generalized in a different direction by Lemari\'e-Rieusset in \cite{LR2} and by Chae and Wolf in \cite{Chae-Wolf}, both working on the whole space.  In \cite{LR2}, Lemari\'e-Rieusset uses the Leray-Schauder approach to first construct self-similar solutions for initial data $v_0$ satisfying $|v_0(x)|\lesssim |x|^{-1}$.  This construction is more general than that in \cite{JiaSverak} but less general than that in \cite{BT1}.  But, Lemari\'e-Rieusset also noticed that provided $v_0$ is self-similar, then $v_0\in L^2_{\loc}$ if and only if $v_0\in L^2_{\mathrm{uloc}}$.  And, furthermore, if $v_0$ is self-similar and belongs to $L^2_{\mathrm{uloc}}$, then it can be approximated by a sequence $v_0^{(k)}$ where each $|v_0^{(k)}(x)|\lesssim |x|^{-1}$. Then, his first construction gives local Leray solutions for each $v_0^{(k)}$ and, because local Leray solutions satisfy the a priori bound \eqref{ineq.apriorilocal} depending only on the $L^2_{\mathrm{uloc}}$ norm of their initial data, these will converge to a SS local Leray solution with $L^2_{\loc}$ data.  Thus, Lemari\'e-Rieusset is able to prove in \cite{LR2} that, for any initial data $v_0\in L^2_{\loc}$ that is incompressible and self-similar, there exists a self-similar local Leray solution evolving from $v_0$. This argument breaks down for DSS solutions since $L^2_{\loc}\cap DSS \neq L^2_{\mathrm{uloc}}\cap DSS$.  Chae and Wolf, on the other hand, use an entirely new method to construct $\lambda$-DSS solutions for any $\lambda>1$ and initial data $v_0\in L^2_{\loc}({\mathbb R }^3)$. These solutions live in the class of ``local Leray solutions with projected pressure,'' which means they satisfy a modified local energy inequality instead of the classical local energy inequality of \cite{CKN}.  We expect this result can be refined slightly; in particular, it seems possible to construct distributional solutions to \eqref{eq:NSE} for any SS/DSS data (or RSS/RDSS data) which satisfy the local energy inequality in the sense of \cite{CKN}. This will be addressed in the future.

\subsection{Next directions}

Theorem \ref{thrm.new} is the strongest available in the scale of critical spaces at the time of writing this survey.  We expect that solutions exist for SS/DSS data in $\dot B_{p,\infty}^{3/p-1}$ for any $3<p<\infty$, that is, the assumption that $p<6$ is likely not needed.  Indeed, using a lemma from \cite{BT3} (see Lemma \ref{lemma.profileslicing} in the next section), it is possible to show that any DSS datum in $\dot B_{p,\infty}^{3/p-1}$ can be approximated by data in $L^3_w$. Each approximate data then gives rise to a DSS solution to \eqref{eq:NSE} by Theorem \ref{thrm:BT1}.  It seems that these solutions \emph{should} converge to a DSS solution to \eqref{eq:NSE} provided some presently unknown a priori bound holds for data in spaces larger than $L^2_{\mathrm{uloc}}$, but we have not found a proof of this.

Although we constructed RSS/RDSS solutions for data in $L^3_w$,
our method for SS/DSS data in $\dot B_{p,\infty}^{3/p-1}$ does not extend to RSS/RDSS data in $\dot B_{p,\infty}^{3/p-1}$.
In particular, we have not been able to prove a counterpart of Lemma \ref{lemma.profileslicing} for such data,
since rotations do not change the size but do change the frequency.
It would be interesting to decide if such a construction is intrinsically possible.

Recall that for small data existence is also known in $BMO^{-1}$.  The $\dot B_{p,\infty}^{3/p-1}$ argument outlined above has no hope of working in $BMO^{-1}$ because, it appears, Lemma \ref{lemma.profileslicing} breaks down for $v_0\in BMO^{-1}$.  This is because $BMO^{-1}$ is an $L^\infty$ based space while $\dot B_{p,\infty}^{3/p-1}$ is an $L^p$ based space ($p<\infty$) -- i.e.~$\|\cdot \|_{BMO^{-1}}$ is computed by taking the supremum over quantities computed on dyadic cubes of (and above) a fixed scale while $\|\cdot \|_{\dot B_{p,\infty}^{3/p-1}}$ is computed by taking the $l^p$ norm over quantities computed on dyadic cubes of a fixed scale (at least, in the wavelet characterization of norms).  In $l^p$, the tail of a convergent series can be made small if $p<\infty$ -- this is the idea behind Lemma \ref{lemma.profileslicing} and clearly fails in an $l^\infty$ based space.
Thus we expect that a totally new approach is needed to address the existence of large data SS/RSS/DSS/RDSS solutions for data in $BMO^{-1}$.

\section{Constructing DSS/SS solutions: a case study}\label{sec.construction}

In this section we will follow the procedure of \cite{BT3} to construct forward SS/DSS solutions with data in the Besov spaces $\dot B_{p,\infty}^{3/p-1}({\mathbb R }^3)$ where $3<p<6$.   In comparison to \cite{BT3} we include an additional step that reveals more about the structure of the solutions -- in particular, the solution satisfies a local energy inequality away from $t=0$.

We will use the following function spaces:
\begin{align*}
&\mathcal V=\{f\in C_0^\infty({ {\mathbb R }^3;{\mathbb R }^3}) ,\, \nabla \cdot f=0 \},
\\& X = \mbox{the closure of~$\mathcal V$~in~$H^1({\mathbb R }^3)$} ,
\\& H = \mbox{the closure of~$\mathcal V$~in~$L^2({\mathbb R }^3)$}.
\end{align*}
Let $X^*({\mathbb R }^3)$ denote the dual space of $X({\mathbb R }^3)$.
Let $(\cdot,\cdot)$ be the $L^2({\mathbb R }^3)$ inner product and $\langle\cdot,\cdot\rangle$ be the dual product for $H^1$ and its dual space $H^{-1}$, or that for $X$ and $X^*$.
Denote by  $\mathcal D_T$ the collection of all smooth divergence free vector fields in ${\mathbb R }^3 \times {\mathbb R }$ which
are time periodic with period $T$ and whose supports are compact in space.

We treat the DSS and SS cases separately, starting with the former.

\subsection{Proof of Theorem \ref{thrm.new}, DSS case}

Elements of this proof are identical to those in \cite{BT3} and some details are omitted here.  The local energy estimate \eqref{ineq.localv} was not considered in \cite{BT3} and is therefore our main focus.

\medskip
\noindent{\bf Step 1: formulation of an auxiliary problem.}  Recall that if $v$ is a DSS solution for some initial data $v_0$, then $u$ defined by
\eqref{ansatz2}-\eqref{variables2}
satisfies \eqref{eq:timeDependentLeray}, i.e.
\EQN{
  \partial_s u
  %
  -  \frac 12 u -  \frac y2\cdot\nabla u - &\Delta_y u +u\cdot\nabla u+\nabla p  = 0,
 \\ & \nabla \cdot u = 0,
}
and, furthermore, $u$ is time periodic with period $T=2\log \lambda$.
Therefore, it suffices to construct $u$ and then show the corresponding vector field $v$ satisfies the properties in the statement of Theorem \ref{thrm.new}.

Let $\epsilon_0>0$ be small enough that there exists a global strong solution to the Navier-Stokes equations for any data in $\dot B_{p,\infty}^{3/p-1}$ with norm smaller than $\epsilon_0$.   In \cite{BT3} we proved the following lemma.

\begin{lemma}[{\cite[Lemmas 2.2, 5.2]{BT3}}]
\label{lemma.profileslicing}
Let $f$ be a $\lambda$-DSS, divergence free vector field in ${\mathbb R }^3$, and belong to $\dot B_{p,\infty}^{3/p-1}$ for some $\lambda \in (1,\infty)$ and $p\in (3,\infty)$. For any $\epsilon>0$, there exist divergence free $\lambda$-DSS distributions  $a\in L^3_w$ and $b\in \dot B_{p,\infty}^{3/p-1}$ so that
 $f = a + b$ and $ \|b  \|_{\dot B_{p,\infty}^{3/p-1}}<\epsilon$.
{If $f$ is self-similar, then so are $a$ and $b$.}
\end{lemma}

Apply Lemma \ref{lemma.profileslicing} for $v_0$ and the prescribed value $\epsilon_0$.  Then, $v_0=a_0+b_0$ where $b_0\in \dot B_{p,\infty}^{3/p-1}$, $\|b_0\|_{\dot B_{p,\infty}^{3/p-1}}<\epsilon_0$, and $a_0\in L^3_w$.  Let $b$ be the strong mild solution to the Navier-Stokes equations for the initial data $b_0$ and let $p_b$ denote its pressure.  By uniqueness in the class of strong mild solutions, $b$ is DSS.  Therefore, there exists an ansatz $B$ for $b$ which is time-periodic with period $T=2\log \lambda$, divergence free, and which satisfies
\EQ{\notag
  \partial_s B
  %
  &-  \frac 12 B -  \frac y2\cdot\nabla B -\Delta_y B +B\cdot\nabla B+\nabla p_B  = 0.
}

Let $A=u-B$. Then, $A$  satisfies
\EQ{
\label{eq:wholeSpaceLeray}
  \partial_s A
  %
  -  \frac 12 A -  \frac y2\cdot\nabla A -\Delta_y A
   +&A\cdot\nabla A + A\cdot\nabla B + B\cdot \nabla A+\nabla p_A  = 0,
   \\ & \div A=0.
}
Letting $a(x,t)$ be the image of $A(y,s)$ under
\eqref{ansatz2}-\eqref{variables2}, we see that $a$ satisfies \eqref{eq.a}
and (formally) converges to $a_0$ as $t\to 0^+$.  Thus, constructing $A$ (in some specific context) will, after changing variables, give us the desired solution $v=a+b$ to the Navier-Stokes equations. To be more specific, we will construct a solution $A$ of \eqref{eq:wholeSpaceLeray}
subjected to the spatial boundary condition
\begin{align}\label{eq:wholeSpaceLeray2}
	\displaystyle \lim_{|y_0|\to\infty} \int_{B_1(y_0)}|A(y,s)-U_0(y,s)|^2\,dy= 0,
\end{align}
and which is $T$-periodic, i.e.
\begin{align}\label{eq:wholeSpaceLeray3}
  A(\cdot,s)=A(\cdot, s+T),
 \end{align}
for a given $T$-periodic divergence free vector fields $B$ and $U_0$.  Here, $B$ is as above and $U_0$ is defined to be
\[
U_0(y,s)=\sqrt t e^{t\Delta}a_0 (x).
\]
Thus, $U_0$ serves as the boundary value of the system for $A$ and encodes information about $a_0$ as boundary data.

It can be shown that $U_0$ satisfies the following assumption (see \cite{BT2}):
\begin{assumption} \label{AU_0}
The vector field $U_0(y,s) :{\mathbb R }^3 \times {\mathbb R } \to {\mathbb R }^3$ is continuously differentiable in $y$ and $s$,  periodic in $s$ with period $T>0$, divergence free, and satisfies
\begin{align*}
& 	\partial_s U_0
-\Delta U_0-\frac 1 2 U_0-\frac 1 2 y\cdot \nabla U_0 = 0,
\\& U_0\in L^\infty (0,T;L^4\cap L^q({\mathbb R }^3)),
\\& \partial_s U_0\in L^\infty(0,T;L_{\mathrm{\loc}}^{6/5}({\mathbb R }^3)),
\end{align*}
and
\[
\sup_{s\in [0,T]}\|U_0  \|_{L^q({\mathbb R }^3\setminus B_R)}\leq \Theta(R),
\]
for some $q\in (3,\infty]$ and $\Theta:{\mathbb R }_+\to {\mathbb R }_+$ such that $\Theta(R)\to 0$ as $R\to\infty$.
\end{assumption}

When we say that $A$ solves \eqref{eq:wholeSpaceLeray}--\eqref{eq:wholeSpaceLeray3}, we mean it is a periodic weak solution in the following sense.

\begin{definition}[Periodic weak solution]
\label{def:periodicweaksolutionR3}
Let $U_0$ satisfy Assumption \ref{AU_0} and assume $B$ is $T$-periodic and divergence free.
The field $A$ is a periodic weak solution to \eqref{eq:wholeSpaceLeray} in ${\mathbb R }^3\times (0,T)$ if it is divergence free, if
\begin{equation}\notag
A-U_0\in L^\infty(0,T;L^2({\mathbb R }^3))\cap L^2(0,T;H^1({\mathbb R }^3)),
 \end{equation}
and if
\EQ{\label{u.eq-weak}
&\int_0^T \bigg( (A,\partial_s f)-(\nabla A,\nabla f)
\bigg)  \,ds
\\& = -\int_0^T \bigg( \frac 1 2 A+\frac 1 2 y\cdot\nabla A-A\cdot\nabla A -A\cdot \nabla B -B\cdot\nabla A,f  \bigg)  \,ds,
}
holds for all $f \in \mathcal D_T$.
This latter condition implies that $A(0)=A(T)$.

\end{definition}

In the above, $A-U_0$ is in the energy class. This hints that we will use energy estimates and compactness arguments to construct $A-U_0$.  Attempting to do so, we hit a roadblock when trying to obtain the formal bound
\[
 \int_0^T\int (A\cdot\nabla U_0)\cdot A \,dy\,ds\leq \gamma \|  A\|_{H^1}^2,
\]
where $\gamma$ is a prescribed, small parameter.  Basically, $U_0$ isn't small and we therefore need to replace it with something that is both small and asymptotically comparable to $U_0$. This is possible because $a_0$ has some decay at spatial infinity.  This allowed us to prove the following lemma in \cite{BT1}.

\begin{lemma}
\label{lemma:W}
Fix $q\in (3,\infty]$ and suppose $U_0$ satisfies Assumption \ref{AU_0} for this $q$.
Fix $Z\in C^\infty({\mathbb R }^3)$ with $0 \le Z \le 1$, $Z(x)=1$ for $|x|>2$ and $Z(x)=0$ for $|x|<1$.
For any $\alpha\in (0,1)$, there exists $R_0=R_0(U_0,\alpha)\ge 1$ so that letting $\xi(y) =Z(\frac y{R_0})$ and setting
\begin{equation}
  W (y,s)= \xi(y) U_0(y,s) + w(y,s),
\end{equation}
where
\begin{equation}
w(y,s)=\int_{{\mathbb R }^3}\nabla_y \frac 1 {4\pi |y-z|} \nabla_z \xi(z) \cdot U_0 (z,s) \,dz,
\end{equation}
we have that $W$ is locally continuously differentiable in $y$ and $s$, $T$-periodic, divergence free,
 $U_0 - W \in L^\infty(0,T; L^2({\mathbb R }^3))$, and
\begin{equation}\label{ineq:Wsmall}
\|W\|_{L^\infty(0,T;L^q({\mathbb R }^3))}\leq \alpha, %
\end{equation}
\begin{equation}\label{WL4.est}
\norm{W}_{L^\infty(0,T;L^4({\mathbb R }^3))}\leq c(R_0,U_0),
\end{equation}
and
\begin{equation}
\label{LW.est}
\norm{	\partial_s W
-\Delta W-\frac 1 2 W-\frac 1 2 y\cdot \nabla W}_{L^\infty(0,T; H^{-1}({\mathbb R }^3))} \leq c(R_0,U_0), %
\end{equation}
where $c(R_0,U_0)$ depends on $R_0$ and quantities associated with $U_0$ which are finite by Assumption \ref{AU_0}.
\end{lemma}
For the proof, see \cite{BT1} (a more technical but more robust proof is given in \cite{BT2} which is also valid on the half-space).

It is now clear that we should seek a solution of the form \[U:=A-W.\]
A bit of calculus reveals that the weak formulation for $U$ is: for all $f \in \mathcal V$ and a.e.~$s\in (0,T)$,
\begin{align}\label{perturbed-Leray}
\frac d {ds}(U,f)
&=
		- (\nabla U,\nabla f)
		+ (%
		\frac 12 U+\frac 12 y\cdot \nabla U, f) - (U \cdot\nabla U, f)
\\
\notag&
		-(W\cdot\nabla U+U\cdot \nabla W + U\cdot \nabla B +B\cdot \nabla U ,f)
\\
\notag& 		-\langle \mathcal{R}(W),f\rangle,
\end{align}
where
\begin{align}\label{RW.def}
\mathcal{R}(W) & :=  	\partial_s W %
-\Delta W
\\&-\frac 12W-\frac 12y\cdot \nabla W +B\cdot\nabla W+W\cdot\nabla B+ W\cdot\nabla W .\notag
\end{align}
In \cite{BT1}, the solution, which evolves from data in $L^3_w$, satisfies the local energy inequality.  It turns out that the same is true of $a$ in the present context, although this was not pursued in \cite{BT3}.  In order to prove this, we modify the formulation of the problem by mollifying the drift velocity.  To this end, for $\epsilon>0$, let $\eta_\epsilon(y)=\epsilon^{-3}\eta(y/\epsilon)$ where $\eta\in C_0^\infty$ is fixed and satisfies $\int_{{\mathbb R }^3}\eta\,dy=1$. The mollified $U$ equation becomes
\begin{align}\label{eq.mollified-perturbed-Leray}
\frac d {ds}(U,f)
&=
		- (\nabla U,\nabla f)
		+ (%
		\frac 12 U+\frac 12 y\cdot \nabla U, f)
		- ((\eta_\epsilon *U) \cdot\nabla U, f)
\\
\notag &  \quad
		-(W\cdot\nabla U+U\cdot \nabla W + U\cdot \nabla B +B\cdot \nabla U ,f)
		-\langle \mathcal{R}(W),f\rangle.
\end{align}

\noindent{\bf Step 2: Auxiliary problem -- approximation.}
We use the Galerkin method as in \cite{BT1}.
Let $\{{a_k}\}_{k\in {\mathbb N}} \subset \mathcal V$ be orthonormal in $L^2_\sigma ({\mathbb R }^3)$ and its span be dense in $X$.
For a fixed $k$, we look for an approximation solution of the form $U_k(y,s)= \sum_{i=1}^k b_{ki}(s)a_i(y)$.
We first prove the existence of and \emph{a priori} bounds for $T$-periodic solutions $b_k=(b_{k1},\ldots,b_{kk})$ to the system of ODEs
\begin{align}\label{eq:ODE}
\frac d {ds} b_{kj} = & \sum_{i=1}^k A_{ij}b_{ki} +\sum_{i,l=1}^k B_{ilj} b_{ki}b_{kl} +C_j,%
\end{align}
for $j\in \{1,\ldots,k\}$,
where
\begin{align}
\notag A_{ij}&=- (\nabla a_{i},\nabla a_j)
		+ (%
		\frac 12 a_i+\frac y2 \cdot \nabla a_i, a_j)
\\\notag &\quad\,	 -(a_i\cdot \nabla (W+B),a_j)
		- ((W+B)\cdot\nabla a_i, a_j)
\\\notag B_{ilj}&=- ((\eta_\epsilon*a_i) \cdot\nabla a_l, a_j)
\\\notag C_j&=-\langle \mathcal R (W),a_j\rangle.
\end{align}

We will show that,
 for any $k\in \mathbb N$ and $\epsilon>0$, the system of ordinary differential equations \eqref{eq:ODE} has a $T$-periodic solution $b_{k}\in H^1(0,T)$.  Letting
\begin{equation} \notag
U_{k,\epsilon}(y,s)=\sum_{i=1}^k b_{ki}(s)a_i(y),
\end{equation}it is possible to show that
\begin{equation}\label{ineq:uniformink}
||U_{k,\epsilon}||_{L^\infty (0,T;L^2({\mathbb R }^3))} + ||U_{k,\epsilon}||_{L^2(0,T;H^1({\mathbb R }^3))}<C,
\end{equation}where $C$ is independent of $k$ and $\epsilon$.   We sketch the proof:  First, we solve the problem \eqref{eq:ODE} for an arbitrary $U^0\in \operatorname{span}(a_1,\ldots, a_k)$
in a possibly short time interval $[0,T']$ (this is a classical ODE problem for which smooth solutions exist) and obtain energy estimates for the solution using Gronwall's inequality.  In particular, by applying the estimates from \cite[(2.18)--(2.22)]{BT2} and \cite[(3.11)--(3.16)]{BT3}, we see that
\begin{equation}
\begin{split}
e^{s/4} ||U_{k,\epsilon}(s)||_{L^2}^2
&\leq ||U^{0}||_{L^2}^2 +  \int_0^{  T} e^{\tau/4}  C_2 \,d\tau
\\
& \le  ||U^{0}||_{L^2}^2 + e^{T/4} C_2 T,
\end{split}
\end{equation}
for all $s\in [0, T']$. Using this inequality, we can extend the existence time to $T'=T$ and choose $\rho>0$ (independent of $k$ and $\epsilon$) so that
\begin{equation}\notag
 ||U^{0}||_{L^2}\leq \rho \Rightarrow ||U_{k,\epsilon}(T')||_{L^2}\leq \rho.
\end{equation}
The mapping $\mathcal T:B_\rho^k\to B_\rho^k$ given by $\mathcal T(b_{k}(0))=b_k(T)$, where $ B_\rho^k$ is the closed ball of radius $\rho$ in ${\mathbb R }^k$, is continuous. Thus $\mathcal T$ has a fixed point by the Brouwer fixed-point theorem, i.e.~there exists some $U^{0}\in \operatorname{span}(a_1,\ldots,a_k)$ so that $b_k(0)=b_k(T)$.  It follows that $U_{k,\epsilon}(T)=U^0(T)=U_{k,\epsilon}(0)$, i.e.~$U_{k,\epsilon}$ is $T$-periodic.
Finally, the uniform bound \eqref{ineq:uniformink} follows from the energy estimate.

\medskip
\noindent{\bf Step 3: Auxiliary problem -- convergence.}  We have two limiting parameters, $\epsilon$ and $k$.  We first take a limit in $k$ to obtain a solution $U_\epsilon$ from the sequence $\{U_{k,\epsilon}\}$. Standard arguments (e.g.~those in \cite{Temam}) imply that, for $T>0$ and for any $\epsilon>0$, there exists $T$-periodic
$U_\epsilon\in {L^2(0,T;H^1({\mathbb R }^3))}$
(with norm bounded independently of $\epsilon$) and a subsequence of $\{U_{k,\epsilon}\}$ (still denoted by $U_{k,\epsilon}$) so that
\begin{align*}
& U_{k,\epsilon}\rightarrow U_\epsilon \mbox{~weakly in}~L^2(0,T;X),
\\& U_{k,\epsilon}\rightarrow U_\epsilon \mbox{~strongly in}~L^2(0,T;L^2(K))  \mbox{~for all compact sets~}K\subset {\mathbb R }^3,
\\& U_{k,\epsilon}(s)\rightarrow U_\epsilon(s) \mbox{~weakly in}~L^2 \mbox{~for all}~s\in [0,T].
\end{align*}
The weak convergence guarantees that $U_\epsilon(0)=U_\epsilon(T)$.  The limit $U_\epsilon$ is a periodic weak solution of the mollified perturbed Leray system.  The variational problem made no mention of the pressure.  Since we want to prove $a$ satisfies the local energy inequality, we need to construct the pressure explicitly.  To do this note that $B\cdot\nabla W +W\cdot\nabla B\in L^\infty(H^{-1})$.  Then, the argument of \cite[Proof of Theorem 2.4]{BT1} applies essentially verbatim and we conclude that there exists $p_\epsilon$ so that $U_\epsilon$ and $p_\epsilon$ are a classical solution to the mollified, $W$- and $B$-perturbed, $T$-periodic Leray equations and, furthermore, $p_\epsilon\in L^{5/3}({\mathbb R }^3\times [0,T])$ with norm bounded uniformly in $\epsilon$.  Because these are classical solutions, they satisfy local energy \emph{equalities}.

We finally let $\epsilon\to 0$.
Because $U_\epsilon$ are bounded independently of $\epsilon$ in \[L^\infty (0,T;L^2({\mathbb R }^3))  \cap   L^2(0,T;H^1({\mathbb R }^3)),\] and $U_\epsilon$ is a weak solution of \eqref{eq.mollified-perturbed-Leray} with $W$ bounded by Lemma \ref{lemma:W}, there exists a vector field $U\in L^\infty (0,T;L^2({\mathbb R }^3))\cap L^2(0,T;H^1({\mathbb R }^3))$ and a sequence $\{ U_{\epsilon_k}\}$ of elements of $\{U_\epsilon \}$ so that
\begin{align*}
& U_{\epsilon_k} \rightarrow U \mbox{~weakly in}~L^2(0,T;X)
\\& U_{\epsilon_k}\rightarrow U \mbox{~strongly in}~L^2(0,T;H(K)) ~ \forall \mbox{~compact sets $K\subset {\mathbb R }^3$}
\\& U_{\epsilon_k}(s)\rightarrow U(s) \mbox{~weakly in}~L^2 \mbox{~for all}~s\in [0,T],
\end{align*}
as $\epsilon_k\to 0$.   Furthermore, since $p_{\epsilon_k}$ are uniformly bounded in $L^{5/3}({\mathbb R }^3\times [0,T])$ we can extract a subsequence (still denoted $p_{\epsilon_k}$) so that
\begin{equation}
p_{\epsilon_k}\rightarrow p_U \mbox{~weakly in}~L^{5/3}({\mathbb R }^3\times [0,T]),
\end{equation}
for some distribution $p_U\in L^{5/3}({\mathbb R }^3\times [0,T])$ and this convergence is strong enough to ensure that $(U,p_U)$ solves \eqref{eq:wholeSpaceLeray} in the distributional sense.

We now prove that $A=U+W$ satisfies a local energy inequality.  Earlier we noted that $A_\epsilon=U_\epsilon+W$ satisfies a local energy equality. Indeed, if $\psi \in \mathcal C_0^\infty({\mathbb R }^4)$ is non-negative, then,
multiplying the $A_\epsilon$-equation by $2A_\epsilon \psi$ and
integrating over ${\mathbb R }^3 \times {\mathbb R }$,
\begin{align*}
\iint &\bke{ 2|\nabla A_\epsilon|^2+\tfrac 12 |A_\epsilon|^2} \psi\,dy\,ds
\\
&= \iint |A_\epsilon|^2(\partial_s \psi +\Delta \psi)\,dy\,ds
\\
&+\iint \bke{ |A_\epsilon|^2 \bkt{ \eta_\epsilon * U_\epsilon+W +B-\tfrac 12y } + 2p_{A_\epsilon} A_\epsilon} \cdot \nabla \psi\,dy\,ds
\\&+ \iint \bkt{ (\eta_\epsilon * U_\epsilon - U_\epsilon)\cdot {\nabla} W - A_\epsilon \cdot\nabla B}\cdot 2A_\epsilon\psi \,dy\,ds,
\end{align*}
where
\[
p_{A_\epsilon} = \sum_{i,j} R_iR_j \big \{
(\eta_\epsilon * U_{\epsilon,i}+W_i)U_{\epsilon,j}
+A_{\epsilon,i}(W_{j}
+ B_j)+B_i A_{\epsilon,j}\big \}.
\]

As in the discussion following \cite[(A.51)]{CKN}, the left hand side of the above equality is lower semi-continuous as $\epsilon\to 0$, while each term on the right-hand side converges to the corresponding $A$ term.   Therefore, after several cancellations, $A$ satisfies
\begin{align}\label{ineq.localA}
&\iint \bke{ 2|\nabla A|^2+\tfrac 12 |A|^2} \psi\,dy\,ds
\\\notag &\leq \iint |A|^2(\partial_s \psi +\Delta \psi)\,dy\,ds
\\\notag
&+\iint \bke{ |A|^2 \bkt{ A +B-\tfrac 12y} + 2p_A A} \cdot \nabla \psi\,dy\,ds
\\\notag &- \iint \bkt{ A \cdot\nabla B}\cdot 2A\psi \,dy\,ds,
\end{align}
where
\[
p_A = \sum_{i,j} R_iR_j \big \{
A_iA_{j}
+A_{i}B_j+B_i A_j\big \}.
\]

\medskip

\noindent{\bf Step 4: Loose ends}

At this point we have constructed a solution $A$ and associated pressure $p_A$ to \eqref{eq:wholeSpaceLeray}.  Since $B$ is prescribed, this gives us a solution $u=A+B$ of \eqref{eq:Leray-rot}, and this corresponds to a solution $v=a+b$ of the Navier-Stokes equations.  Except for the local energy inequality \eqref{ineq.localv}, all the conclusions of the theorem follow identically to \cite[Proof of Theorem 1.4]{BT3}.

For the local energy inequality \eqref{ineq.localv}, we first show that \eqref{ineq.localA} for $A,p_A$ is equivalent to
the following $b$-perturbed local energy inequality for $a$ and $p_a$: if $\phi\in C^\infty_0({\mathbb R }^3\times {\mathbb R }^3_+)$ is non-negative then
\EQ{ \label{ineq.locala}
&2\iint |\nabla a|^2\phi\,dx\,dt
\leq  \iint |a|^2(\partial_t \phi + \Delta\phi )\,dx\,dt %
\\&\quad + \iint [|a|^2 (a+b) + 2p_a a]
\cdot\nabla \phi\,dx\,dt - 2\iint (a\cdot\nabla b) \cdot (a\phi)\,dx\,dt.
}
To see this, let $\phi(x,t)=\frac 1 {\sqrt t} \psi(y,s)$, where $\psi$ is a test function from \eqref{ineq.localA}. Using the relationships between $a$, $A$, $b$, $B$, $p_a$, $p_A$, $y$, $x$, $s$, and $t$, it is possible (after a lengthy computation) to derive \eqref{ineq.locala} directly from \eqref{ineq.localA}.  The other direction follows from the same computation.

To show \eqref{ineq.localv}, note that $b$ is smooth for $t>0$, and the identities from integration by parts of
\[
\iint ({\partial}_t a - \Delta a + (a+b)\cdot\nabla  a + a \cdot \nabla b + \nabla p_a)\cdot 2b \phi=0
\]
and
\[
\iint ({\partial}_t b - \Delta b + b\cdot \nabla b+\nabla p_b)\cdot 2v \phi=0
\]
are valid. The sum of these identities and \eqref{ineq.locala} gives  \eqref{ineq.localv}.

This completes the proof of Theorem \ref{thrm.new} in the DSS case.

\subsection{Proof of Theorem \ref{thrm.new}, SS case}

The SS case can be proved directly by constructing stationary  solutions to \eqref{eq:stationaryLeray}.  Alternatively, they can also be obtained as a limit of DSS solutions.  We give a sketch of the proof of Theorem \ref{thrm.new} using the limit approach since it illustrates the usefulness of the a priori bound \eqref{ineq.apriorilocal} satisfied by local Leray solutions.  This approach was also used in \cite[Proof of Theorem 1.3]{BT1} to obtain SS solutions as a limit of DSS solutions.

\begin{proof}%

Assume $v_0$ is SS and belongs to $\dot B_{p,\infty}^{3/p-1}$.  Let $a_0$ and $b_0$ come from Lemma \ref{lemma.profileslicing} for a small constant $\epsilon_0$ (smaller than the Koch-Tataru constant) and let $b$ be the unique mild solution to the Navier-Stokes equations \eqref{eq:NSE} with initial data $b_0$.  Let $a_0^{(k)}=a_0$ where $k\in {\mathbb N}$.  Clearly, $a_0^{(k)}$ is $\lambda_k$-DSS with $\lambda_k=2^{2^{-k}}$.  Thus, by the DSS case, there exists a $\lambda_k$-DSS solution $v^{(k)}=a^{(k)}+b$ to the Navier-Stokes equations.  Moreover, each $a^{(k)}$  satisfies \eqref{ineq.apriorilocal} but where the constant $\sigma$ is modified (in comparison to \eqref{ineq.apriorilocalconstant}) to also depend on $b$ (we omit a proof of this but it follows from the fact that $a$ satisfies a linear perturbation of the Navier-Stokes equations and the argument from \cite{JiaSverak}).  Since $b$ is independent of $k$, this gives a uniform bound on the sequence $a^{(k)}$ and allows us to apply a convergence argument to obtain a SS solution to the Navier-Stokes equations satisfying the properties listed in the statement of Theorem \ref{thrm.new} -- see \cite[Proof of Theorem 1.3]{BT1} for details.
\end{proof}

\section*{Acknowledgments}
The research of both authors was partially supported by the NSERC grant 261356-13 (Canada). That of Z.B. was also partially supported by the NSERC grant 251124-12.

redpajama_set_name: RedPajamaArXiv


text:
Unite the Right's "Swastika Pin" Identified as Zachary Hudson Fisher, aka Milwaukee DJ "Hdsn Acid"
(Note: this post has been updated)
The 2017 "Unite the Right" rally brought hundreds of violent neo-Nazis and other extreme-right elements converging on the city of Charlottesville with intent to do harm to counter-protestors who opposed their message of white supremacy, anti-semitism and hatred of LGBTQ communities. Many of these individuals have since been identified, forcing them to face public scrutiny for their actions, yet some seem to have slipped through the cracks. One of these individuals is a person whom we refer to as "Swastika Pin," due to the enamel Nazi pin he wore on his shirt at the rally.
"Swastika Pin" was one of the more violent participants in the rally and he has eluded identification for three years, but now we have uncovered his identity. In this article we will provide the background of this individual's neo-Nazi activities as well has his true identity and current activities.
"Swastika Pin"
"Swastika Pin" was a very visible participant in the the "Unite the Right" rally and was prominent in the Vice News production "Race and Terror" where he could be seen marching with a crowd of neo-Nazis wielding torches and chanting "Jews will not replace us" on August 11, 2017.
Still from the Vice News segment "Race and Terror." "Swastika Pin" in white shirt.
"Swastika Pin" (in yellow border).
Activist and data scientist Emily Gorcenski came face-to-face with his hatred as he shouted inches from her face and assaulted her and others. She captured the incident on video and posted it here. We have clipped a portion of her video for viewing here (note: video may be unsupported in mobile browsers):
https://accollective.noblogs.org/files/2020/09/cville-swaspin.m4v
Alarmingly, there is evidence that "Swastika Pin" planned to commit acts of violence at the rally as indicated by these screen shots from organizer Jason Kessler's live stream:
Screen shots of comments from "Unite the Right" organizer Jason Kessler's live stream, "Swastika Pin" writing as "Hudson Vandal."
Another screenshot of a comment made by "Swastika Pin" (writing as "Hüdson La'Beauxl") on Facebook also indicates that he intended to commit violence upon Emily Gorcenski, this time implying that he would kill her.
"Swastika Pin" was also prominent on the next day of the rally, August 12.
"Swastika Pin" in yellow border.
"Swastika Pin" bordered in yellow. Note "Mosin-Nagant" (Dallas JN Medina) in black on the right.
"Swastika Pin" wasn't shy about his violence that day either, as evident by this interaction with neo-Nazi Christopher Cantwell:
— Jack Smith IV (@JackSmithIV) August 13, 2017
Emily Gorcenski had done significant research on this individual and found several clues to his identity that would prove to be critical later, such as the fact that "Swastika Pin" was likely a Florida resident and likely had ties to the city of Winter Haven.
In fact, it was Emily Gorcenski who coined the moniker "Swastika Pin," but he has used a variety of aliases across the internet, such as "Hudson Vandal," "James Hudson Vandal" and variations of those names, as well as variations of "Kovalev." In the Discord leaks provided by Unicorn Riot he can be found using the handle "subhuman" as he chatted with fellow neo-Nazis. In one of his early posts "subhuman" points out that he appears in the Vice News segment.
In the invite-only Bowl Patrol Discord chat fellow neo-Nazis, such as "Tactical Bowlcut" (Robert Kehne Moeller) and "Vic Mackey" (Andrew Richard Casarez), members of the "Bowl Patrol," even altered images of him, placing bowl-style haircut images on his head along with a headband that reads "kill Jews," a sign of his acceptance into the Bowl Patrol circle.
Image posted on Discord by "Vic Mackey" (Andrew Casarez) altered with bowl cut and "kill Jews" headband placed on "Swastika Pin's" head. [link]
The Bowl Patrol, as we have written about in earlier articles, is a loose organization of neo-Nazis who admire the racially motivated mass murderer Dylann Roof.
Another altered photo, this one by "Tactical Bowlcut" (Robert Kehne Moeller). [link] Note the tattoos.
Photo altered with bowl cuts by "Vic Mackey"/Andrew Casarez, posted on Discord [link]. "Swastika Pin," center; "Tactical Bowlcut"/Robert Kehne Moeller, seated; James Michael Gallagher, right.
"Swastika Pin" also seemed rather good friends with a racist named James Michael Gallagher. Here are the two together in a photo taken by Moeller in a Charlottesville hotel room arranging bowl cuts.
More on Gallagher may be found here.
Gallagher and our "Swastika Pin."
Discord user "Erika" (Erica Joy Alduino) posted a photo in which "Swastika Pin" participated in a demonstration with other well-known neo-Nazi figures. This took place in front of the Pulse nightclub in Orlando, Florida where a mass shooting had occurred nearly year prior.
At some point "Swastika Pin" had visited neo-Nazi William Fears ("Antagonizer") in Austin, Texas.
Will Fears in foreground. "Swastika Pin" in background performing a Nazi salute. [link]
We know the two were fast friends from a post on William Fears' Instagram account that shows them posing together, along with other neo-Nazis in a fight gym.
Instagram post from account belonging to William Fears (grey shirt). "Swastika Pin" in a bright blue shirt. Another known neo-Nazi, Joffre Cross, is in a black shirt.
Much of "subhuman"'s discussion in the Discord leaks had to do with a white supremacist who wished to leave the movement and decided to go public. This person provided an "Ask Me Anything" post on Reddit and provided a link to a video in which members of his group had gathered and chanted "gas the k*kes." "subhuman" states that he was in this video and states as well that he started the chant.
Examining the video, we can indeed find "Swastika Pin" or "subhuman" present.
The distinctive profile of "Swastika Pin" is immediately identifiable.
Cropped still from the Vice News segment "Race and Terror."
Still from the Vice News segment "Race and Terror."
Cropped photo from a Facebook post.
Aside from private Nazi chat rooms, "Swastika Pin" also used aliases on conventional social media platforms to spread hateful memes and rhetoric. Here he is on Facebook as "Hüdson La'Beauxl" (presumably pronounced "bowl") posting anti-Semitic sentiment:
On another Facebook account, "Swastika Pin" poses with neo-Nazi podcaster "Mike Enoch" (Mike Peinovich).
In a Twitter post he mocked the victims of the Pulse nightclub shooting on the one-year anniversary of the event. He was also here two days earlier at the banner event mentioned earlier.
In another post "Swastika Pin" does the "white power" hand signal while kneeling next to a "RAHOWA" bumper sticker. For those unaware, "RAHOWA" stands for "Racial Holy War."
He also provided music for the right-wing podcast "Paranormies" as "Hudson Vandal."
Could it be that "Swastika Pin" might have some musical talent or skills? We'll have our answer later in this article.
Following the 2017 Charlottesville rally, "Swastika Pin" kept a low profile, perhaps fearful that he would be held accountable for his actions during the event. We found this Facebook profile in August of 2019. In this account, "Swastika Pin," now going by the alias "Hudsen Zuccsby," posted a photo of himself sporting a bowl-style haircut, an indication that he still was still a fan of the murderer Dylann Roof. This account was Facebook "friends" with an account used by James Michael Gallagher, indicating that they were still friends during this time.
We could tell from his past behavior online that "Swastika Pin" was comfortable with using variations of familiar user names across his social media accounts. One social media account, however, proved to be especially fateful for this neo-Nazi-in-hiding. "Swastika Pin" had an Instagram account under the name "hudsonacid" (archive) on which he posted several images of himself and, aware of his user name preferences, we searched for variations of this user name, which led us to another Instagram account, one that would lead us to the eventual discovery of "Swastika Pin"'s true identity.
Instagram account "hudsonacid"
HDSN ACID
The Instagram account "hdsn_acid" (archive) was immediately recognizable as belonging to our target since he posted many images and videos of himself, many of which display his distinctive collection of tattoos. After locating his new Instagram account we located his new Facebook account (archive).
Instagram post from "hdsn_acid" Note the tattoos on chest and middle finger.
Facebook post from "HDSN ACID" Note torso tattoos on display.
From the information on this account we learned that "Swastika Pin" had relocated from his native Florida to Milwaukee, Wisconsin and had become a music DJ and composer of electronic dance music. Calling himself "Hdsn Acid," our "Swastika Pin" had performed at a number of festivals and was garnering a following among EDM afficionados. In fact, some of his musical colleagues were persons of color and likely unaware of his violent racist past. It appeared that "Swastika Pin" was determined to appear to be anything other than the violent neo-Nazi that he was.
Post on Facebook by a musical colleague.
"Hdsn Acid" posing with fans on Instagram.
In August "Hdsn Acid" celebrated the release of a single entitled "Can U Hear Me Now?" on the music platform Soundcloud, as well as commercial internet music markets like iTunes and Spotify. He also announced a new EP album entitled "Libra" due for release on October 2, 2020 in the bio portion of his current Instagram page.
We now knew what "Swastika Pin" had been up to recently and we knew his location as well. What we did not know was his full legal name, but this was not far off.
Having a look at Spotify's information on Hdsn Acid's single provided us with the songwriting credits for this song: "Zachary Fisher." Knowing he was likely from Florida we searched the state's public records and found our man: Zachary Hudson Fisher of Winter Haven, Florida.
ZACHARY HUDSON FISHER
The violent neo-Nazi whom we have come to know as "Swastika Pin" was, in fact, Zachary Hudson Fisher, born October 2, 1989 from Winter Haven, Florida. We had postulated that "Swastika Pin" was astrologically a Libra, since he has a Libra tattoo on his stomach, and thought it possible that the release of his album, October 2, might be his birthday. Our findings confirmed that this analysis was correct. Furthermore, "Swastika Pin" followed his modus operandi of repurposing familiar names, in this case his middle name "Hudson."
Looking back through his history on the internet we see that Fisher had experience in amateur music production and a few minor run-ins with the law.
Fisher's mugshot and arrest details found on a mugshot website.
Zachary Hudson Fisher's profile on beatstars.com
Zachary Hudson Fisher as "Fisher on the Beat" on reverbnation.com
Zachary Hudson Fisher's Myspace page.
How Zachary Hudson Fisher initially got involved in white supremacist movements is beyond the scope of this article, but we feel that in light of the violence he demonstrated at the "Unite the Right" rally, his hateful online rhetoric and the fact that he seems to think that he has escaped the consequences of his past actions leads us to believe that he is still a danger to our communities.
d.o.b. 10/2/1989
From Winter Haven, FL; Currently in Milwaukee, WI
Currently DJs under the name "Hdsn Acid"
Many thanks to our friends @ Utah161 on Twitter and the many anonymous comrades who have contributed to this research either directly or indirectly.
Update, 30 September 2020
Fisher has issued public statements on his Facebook page regarding our findings which may be found here [archive] and here [archive]. While we at the Anonymous Comrades Collective believe that everyone is valuable and has the ability to change for the better, we find Fisher's statement to be insufficient given the level of violence he showed at the "Unite the Right" rally and considering how immersed he was in white supremacist movements.
For one, Fisher seems to be expecting instant redemption despite not putting in the work to show that he has changed. Secondly, Fisher is shifting blame, citing his drug abuse and claiming that he was a "victim" of violence at the rally, a rally that at which he and other white supremacists intended to instigate violence. Thirdly, he and his defenders have made claims that aspects of our research had been fabricated. This is untrue. To be clear, the only alterations of any photos in this article were 1) markup to point out the subject of the photo in context and 2) cropping and/or redactions to protect privacy.
Fisher cites some of the altered or "photoshopped" photos that include a "bowl cut" in them; these are altered photos and we believe we have sufficiently explained who did these and why in the article. He also claims the threats of violence are faked:
This is untrue. They are unaltered screenshots of statements he made on publicly visible social media.
Fisher's significant other of "5 years" has been vocal in defending him (although her statements are public we will not identify her here). She has brought forth suspicion that some of Fisher's social media accounts have been "faked."
We are not aware of any fake Instagram or Soundcloud pages attributed to Fisher. All of the accounts cited in the article, to the best of our knowledge, belong to Zachary Hudson Fisher.
Again, we hope that Fisher will one day have the self-awareness to enable true personal growth, but it is evident by the response to his exposure that he is not willing to take full responsibility for himself. There indeed have been individuals who have made tremendous efforts to escape hateful ideologies. Fisher's attempt to talk his way out of his past is an insult to those admirable individuals who have truly left the hate.
Zachary Hudson Fisher (left) side-by-side with white supremacist leader Evan Mclaren (right) at the "Unite the Right" rally in 2017. Fisher's proximity to such high-profile individuals in white supremacist movements is indicative of how deeply he was immersed in hateful ideologies.
Zachary Hudson Fisher posing with rifle.
Posted byancoco 2020-09-24 2020-09-30 Posted inGeneralTags: Andrew Casarez, Bowl Patrol, Charlottesville, Hdsn, Hdsn Acid, Hudson Vandal, Robert Kehne Moeller, Swastika Pin, Tactical Bowlcut, Vic Mackey, Will Fears, Zachary Hudson Fisher
Doxer Gets Doxed: "Proud Boy" Chadwick Jason Seagraves of NCSU

redpajama_set_name: RedPajamaCommonCrawl


Here's the interesting part. I believe the color pie is the heart of the game. I think it's at the core of what makes Magic tick. That's as far from "ruining the game" as you can get. So I thought I'd dedicate my column today to explaining in detail why the color pie is so important. Hopefully, this will allow many of you to see it with the same appreciation I do.
Let's get to it. What's so great about the color pie?
I started with this one because I think many of the color pie's detractors consider the restrictions created by the color pie to one of the biggest strikes against it. After all, restrictions keep players from doing the things they want. That's bad, right? Well, not to a game designer. You see, it's our job to make your life difficult.
Overcoming obstacles is the joy of strategy gaming. ("Oh my god, he got me down to one life and he had four creatures in play and all I had was a single mountain." "So, you lost?" "No, I won. You're not going to believe how it happened.") Thus, restrictions while frustrating short term, are critical to happiness long term.
"But wait a minute," many of you are saying, "I have an opponent. He's the obstacle. I don't need the game getting in my way." I would argue that you do. Strategy gamers want to be mentally challenged. To do that, the game designers have to make problem-solving hard. Otherwise there isn't enough challenge. A key element to accomplishing this is to limit the tools available to solve the task.
To explain, let me pull a story from my life. My sophomore year in college I joined the student government of my dorm. As such, I was expected to participate in numerous "spirit-building" activities. One such activity was building a dorm float for the homecoming parade. Now, the dorm government wasn't all that thrilled with the idea so they put very little money towards the project. A hundred dollars if I remember correctly. Other dorms were spending over a thousand dollars. We had a hundred.
I showed up around nine o'clock (the floats were built during an all-nighter the night before the big game) to discover that the entire team was six people. Most other floats had fifty or more people working on them. We had twelve hours, an eighth of the people and a tenth of the money to build a float. So we lost? No, we won, and you're not going to believe how it happened. Somehow during the night we came up with a unique idea that played into our lack of resources. We found inventive ways to aid in our building and pieced together a rather impressive float. I had a blast.
When I showed up the next year, I discovered that our dorm had joined forces with another dorm. As such we poured tons of money and people into the project. We had every resource at our fingertips. You know what? I hated it. I'm a strategy gamer. I don't want everything just given to me. I want to earn my victories.
This is why the color pie is so important. As a game designer, I want to make the players earn their victories. The game simply wouldn't be as fun if every color had access to every effect. Each color has built in weaknesses that the opponent can exploit. But that doesn't mean you can't find creative solutions. In fact, finding creative solutions is what Magic is all about.
R&D hates flavor… or so the myth goes. I think this myth comes from a few articles where I talked about R&D siding with function when it and flavor clashed. From that, some players jumped to the conclusion that R&D finds function more important. I guess hating flavor is then only a hop, skip and a jump away.
R&D doesn't hate flavor. In fact, I think the color pie is proof of how much R&D appreciates flavor. As I said above, I believe the color pie is at the core of the game. And the color pie is an example of a case where flavor trumps function. It's actually a bit more severe than that. In the color pie, flavor dictates function. Flavor tells function what it can and cannot do.
This is one of the most important roles of the color pie. It clarifies each color's philosophy and then extends them to the game by bending mechanics around it. One of the reasons that the colors are so rich in Magic is that they extend to every aspect of the game.
The Color Wheel is a chart that explains the flavor and conflicts of each color, and it may or may not be synonymous with the Color Pie. It's all very confusing. And no, we have never put a large version of this document up on the web.
Another important goal of the game designer (and more importantly the game developer) is to keep the game balanced. One of the defining qualities of Magic deck design is that no strategy is unbeatable. Every archetype has its weakness. If a player knows what his opponent is playing, he should be able to defeat it.
In addition, each color has a certain style of play that causes it problems. Red, for example, has no long-term card advantage. It has strengths in the early game but tends to run of out gas in the later game. Once again, the designers and developers harness weakness as a tool to craft the game. No color can run amok because the game has built-in safety features. In fact, the times that R&D has had the greatest problems is when we allow colors to get access to things they're not supposed to have. When R&D doesn't respect the color pie in always comes back to bite us.
Normally I make my case by talking about the topic. For a change, I thought it might be nice to show you what I mean. You want to experience a world without a color pie? Here's what you do. Build some decks with the following assumption: all basic lands produce any color of mana. This will give you a glimpse into a pie-less world. Go play a whole bunch of games and come back. Don't worry, I'll wait.
Back already? Well, how were the games? At first, they're fun because you're doing something new that you don't normally get to do. But after a while it becomes a little monotonous. Why? First, card variety goes down as each deck just uses the most efficient card available. Soon deck archetypes start collapsing together. There is no white weenie or green "Stompy" deck, just a speed creature deck. Next, the decks start to seem less connected because the cards in them don't have any common theme. What you end up with is a few, very efficient but flavorless decks. You have efficiency at the sake of personality.
At its heart that is the most important part of the color pie. Yes, Magic could be played with one color, but it would be less of a game. Having five colors means that each color has to have its own personality. For that to happen, the colors need a distinct feel both in how the cards look and sound and in how the cards play.
Too often, I think people see the color pie as a tool to lessen what the colors can do. I see the color pie as a tool that highlights what each color represents. It doesn't detract but rather adds to the value of the colors. The more different black is from green, the better. I don't want colors that are all subtle shades of one another. I want colors that are blaringly unique in what they represent. This is what the color pie adds to the game.
As you can see, I'm very passionate about the color pie. I believe Magic owes much of its uniqueness to it. Hopefully, my column today gave you a better insight into R&D's dedication to it. As you can see, there is much more to it than one might see at first glance.
Join me next week when I introduce some unique individuals.
Until then, may you know the importance of being different.


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BAW, Military Vehicles, Post-war,
Beijing BJ212
The Chinese Beijing BJ212 (Chinese: 北京212) and BJ2020 is a Land Rover and Jeep inspired four-wheel drive, diesel-powered light-duty offroad utility vehicle in the half tonne class, built by Beijing Automobile Works. The original design, depending in large on Russian off-roaders from UAZ and possibly developed in conjunction with the UAZ-469, debuted in 1965 as the Beijing BJ212. It has been known as the BJ2020 series since the Chinese car classification system was changed in 1989. It is commonly used by the Chinese government (particularly the armed forces), but is also commercially available. Various versions of the Jeep are today sold under the names of Zhanqi, Jinxuanfeng, Lieying, City Cruiser, Kuangchao, and Ludi. The BJ212 and its successors has a long history of use by the People's Liberation Army.
Retrieved from "https://classiccars.fandom.com/wiki/Beijing_BJ212?oldid=62086"


Inspired by traditional hotel room keys, the Maison key holder pays tribute to LV flagship stores around the world. A Monogram Flower key charm and key ring hang delicately from the snap hook. The Maison oval is engraved with the Flagship 101 Champs-Elysees address. The chain can be clipped on most metallic links on handbags or used as a key holder. What a fabulous way to jazz up your bag! Retail price is $390.
The exterior hardware is clean and beautiful with some minor hairline surface scratches. Overall it is in excellent condition.


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Daniel Grassl – pattinatore artistico su ghiaccio italiano
 Gabriella Grassl – ex sciatrice alpina svedese
 Paulina Grassl – ex sciatrice alpina svedese

redpajama_set_name: RedPajamaWikipedia


1. 모유수유 성공을 위하여 모든 의료인들과 잘 대화되도록 통상적으로 명시된 지침서를 가지시오 Have a written breastfeeding policy that is routinely communicated to all health care staff.
2. 모든 의료인들은 위에 명시된 데로 이행하도록 모유수유를 이해하고 훈련해야 된다. Train all health care staff in skills necessary to implement this policy.
3. 모든 임산부는 모유의 유익성을 알고 모유수유를 관리할수 있도록 정보를 주어야 된다. Inform all pregnant women about the benefits and management of breastfeeding.
5. 엄마가 비록 아기와 떨어져 있어도 어떻게 젖주며 어떻게 수유를 지탱할 수 있는지 보여 주시오 Show mothers how to breastfeed, and how to maintain lactation even if they should be separated from their infants.
6. 어떤 특별한 이유가 없는 한 신생아 에게 모유외 엔 아무것도 주지 마시오.
7. 아기와 엄마를 24시간 함께 하게 하시오 Practice rooming-in - that is, allow mothers and infants to remain together - 24 hours a day.
8. 특정한 시간이 아니라 아기가 원할땐 언제나 젖주시오. Encourage breastfeeding on demand.
10. 엄마를 병원이나 클리닠에서 내어 보낼 때 모유수유지지 그룹에 연결 시켜주고 엄마를 그곳에 부탁해 주시오 Foster the establishment of breastfeeding support groups and refer mothers to them on discharge from the hospital or clinic.
왜 모유가 엄마와 아기를 위해 문제들을 야기하는 인공우유 보다 가장 좋은지 배우시오 .
Breast is best, but why?
2. 세상에 있는 모든 사람들에게 말하시오!
산과의사, 가정, 친구와 동료들에게 모유 먹일계획을 말하고 도움과 지지를 받으시오. 분만실 의사나 간호사 영아실 그리고 산과병동의 간호사에게 당신의 아기는 모유만 먹인다고 알리시오.
Tell the world ! Let your OB doctor, family, friends, and employers know that you are planning to give only breast milk, and need their full support. Tell the Labor &Delivery doctors and nurses as well as the nursery and postpartum nurses that you want only breastmilk for your baby.
3. 출산직후 아기를 피부와 피부를 대고 안아주시오.
아기는 무척 깨어있고 젖 빨려고 대기 중 입니다. 만약 이런 루틴이 아니래두 이것을 원한다고 말하시오.
Hold your newborn skin- to- skin right after birth. Your baby will be alert and interested in breastfeeding. Tell your doctor and nurses that you want this if it is not offered.
그래야 아기의 젖먹고 싶은 눈치를 알수있고 곧 아기에게 응 할 수 있습니다. 아기가 울때까지 기다리지 마시오. 만약 이것이 병원 루틴이 아니래도 말하시오.
Rooming-in with your baby.This will allow you to respond to your baby's cues and breastfeed as soon as he shows he is hungry. Don't wait for him to cry.Ask for rooming -in if it is not offered.
그저 주는것이고 또 옆에 있다고 인공우유를 쓰지 마시오. 어떤 병원은 내과적인 이유가 없는 모든 엄마에게 모두 이 인공 우유를 주고 이 인공우유는 아기가 배가 불러져 엄마 젖을 자주 못 빨게 합니다.
Frequent feeds, not formular. Don't use formular just because it is provided. Some hospitals provid formular toallmothers even if there is nomendicalreason. If formular is given, your baby will be too full to breastfeed often.
직장으로 돌아 가게 될 때 아기가 필요 할 만큼 젖을 내기 위하여 지금 아기에게 젖 만을 주시오. 병을 주는 일은 뒤로 미루시오.
If you will return to work, start off with breastfeeding only so your body makes the milk your baby will need. Introduce bottles later.
7. 만약에 도움이 필요 하시다면 꼭 청하시오.
당신의 간호사에게 모유수유 전문가를 있는가고 물어보시오.
8. 어려움이 있어도 낙심치 마시오.
대개의 문제는 쉽게 고칠 수 있습니다. 모유수유 한 친구나 가족이나 WIC 상담자에게 도움을 청하시오. 또 모유수유 디렉토리에 있는명단을 보시오.
Don't get discouraged if you have difficulties. Most breastfeeding problems can be easily fixed. Askfor help from a friend or relative who has breastfed or your WIC counselor. The breastfeeding Resource Directroy lists breastfeeding help available in your area.
학교나 직장을 위하여 집을 떠나서도 편안하게 모유수유할 수 있는 유축기 사용 계획입니다.
Plan for long term.Get comfortable breastfeeding away from home;make a planfor pumping when you return to work or school.
10. 모유수유 하는 다른 엄마와 교제 하시오. 새로운 친구들을 만들고 라리체 라는 엄마구룹에 합세 해요. 무엇보다 가장 중요한건 아기를 아는 즐거움 입니다.
Get to know other breastfeeding moms:make new friends;join a LaLeache League mother's group. And most importantly, have fun getting to know your baby!


Smart Poly™ material has been designed for charts in the classroom and home. These charts are strong, durable, water resistant, and do not need lamination. They can be displayed like any chart with Ashley Smart Tak™, pins, tacks, staples, etc. The surface is designed for dry erase and water based markers and washable crayons. Chart measures 13" x 19" with a dry-erase, glossy surface, .45mm. The Spanish Emoji ¿Cómo te sientes hoy? (How Are You Feeling Today?) chart features boxes to mark next to each emotion to help children to show how they are feeling.


Brian Moran has spent the last 20 plus years helping America's entrepreneurs realize their dreams. Prior to starting his 3rd company in 2012, he was the Executive Director of Sales Development at the Wall Street Journal, overseeing the financial and small business markets across the WSJ franchise. From 2002-2010, Brian ran Veracle Media & Moran Media Group. Both content companies utilized various media channels to help entrepreneurs manage & grow their companies. Prior to launching these two companies, he was the Associate Publisher at Inc. Magazine, a Publisher and Associate Publisher at Entrepreneur Media and held management positions at Success Magazine and Medical Economics Magazine. He uses this experience to help others grow. Brian has assembled an impressive network of entrepreneurial experts devoted to helping marketers connect to the SMB market. Brian is also a member of the Small Business Entrepreneurship Council, US Chamber of Commerce Small Business Council, and SCORE's National Marketing Council (from which he won the Lou Campanelli Award). Today, Brian discusses politics and the upcoming Presidential election.
Andrew Bermudez has over 12 years of hands-on experience in the commercial real estate brokerage field. He's developed online applications that help automate complex and repetitive processes, while maintaing accuracy and making them very easy to work with. He recently co-founded an internet technology platform that helps convert commercial real estate vacancies, renters and buyers into real estate dollars 17 times faster. His new venture was rated an outstanding and top-scoring graduate of OCTANe's LaunchPad program. Digsy is an on-demand commercial real estate brokerage service. Their technology platform helps business owners save time finding the perfect Office, Warehouse or Retail space for their business with the help of trusted local brokers — free of charge. All within just a few clicks. Their commercial real estate experts devote 100% of their time finding your dream commercial space while you focus on growing your business.


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hollywood collectables
#514 - "The Fam" - April 8, 2019
This week on the show, more casting for Jason Reitman's Ghostbusters confirmed, a new 35th anniversary home video release, and IDW one-shot talk! First up, the boys catch up after last week's April Fool's Yes Have Some episode. Then, Sony Pictures Home Entertainment has announced a UHD 4K release of the original two Ghostbusters films in celebration of the anniversary packed with special features including the Fort Detmerring scene and a few other surprises (including a commentary by this ol' podcast!). Next up, some casting confirmation for "the fam" as McKenna Grace has been confirmed to join Carrie Coon and Finn Wolfhard in a family with mysterious ties to the original Ghostbusters film. In the last section of the show, we'll talk about IDW Comics' excellent 35th Anniversary one-shot. All that, plus your voicemails and a few merch updates, and more in a jam-packed episode of the Crossrip!
Tagged: jason reitman, mckenna grace, finn wolfhard, carrie coon, ghostbusters 35th anniversary, uhd 4k, idw comics, gb35, gb20, ghostbusters (2020), hollywood collectables
#507 - "Ectotron" - February 18, 2019
Hide your wallets, it's International Toy Fair time! This week on the show, the latest Ghostbusters (2020) news, and the late-breaking announcements we were able to talk about from Toy Fair in time for recording! First up, some discussion about an unconfirmed "exclusive" scoop about Ghostbusters (2020)'s production start date. Is there merit to it? Is there room for doubt? In fact, are we taking too many things at face value at the moment? The boys have an in-depth conversation (that's awfully similar to one had three years ago at this point in the production timeline). The Ghostbusters social media account caused quite a stir, Troy and Chris will tell you why. And then some talk about the show-stopping reveal that's more than meets the eye - an Ecto-1/Transformer has become reality.
Tagged: rubies costumes, ghostbusters 3, hollywood collectables, international toy fair, new york toy fair, 35th anniversary, ectotron, jason reitman, toy fair, rubies, ghostbusters (2020), transformers, ernie hudson, ecto-1
#429 - "SDCC 2018 and Remembering Terry Windell" - July 30, 2018
This week on the show, rounding up all of the SDCC 2018 news and announcements and remembering special effects pioneer Terry Windell. First up, out of SDCC there were a ton of new merch announcements, including Spirt Halloween props and items, new 1:1 replicas, action figures and more (including a Slimer Chia Pet?). Then we'll discuss Ghostbusters World, which was the talk of the con for Ghostbusters fans. Many got a hands-on demo of the game and provided their thoughts. On the press tour for Mission Impossible, writer/director Drew Pearce was hit up (again) about his work on Ghostbusters that was abandoned, and had some interesting comments. Finally, we're remembering Terry Windell of ILM and Boss Films who passed last week at the age of 61, the pioneer who was known for his memorable lightning effects in Ghostbusters and Return of the Jedi left a lasting impression that Troy and Chris discuss. All that and more on this week's Crossrip!
Tagged: hollywood collectables, prop, video games, drew pearce, ilm, zuul, boss films, mobile games, sdcc, terry windell, slimer chia pet, ghostbusters (2016), ghostbusters world, spirt halloween, return of the jedi, San Diego Comic-Con, san diego comic-con, special effects, prop replicas
#423 - "Ghostbusters Crossing Over #3" - June 18, 2018
Welcome back to our regularly scheduled Monday programming! This week on the show: Ghostbusters Day clean-up, news and IDW Ghostbusters Crossing Over #3 discussion! First up, some of the items that we didn't touch upon in last week's Ghostbusters Day celebration show, including Diamond Select Toys announcements and photos, the VOID expanding to 20 more cities, Hollywood Collectables' Egon Spengler statue release, and some news on the San Diego Comic-Con fan event. Then in the second half of the show, join us for setting up our Ghostbusters Daughter podcast book club and find out how you can be involved. After that, some spoiler-filled discussion of IDW Comics' latest release in the Ghostbusters event, Crossing Over issue 3.
Tagged: Erik burnham, hollywood collectables, San Diego Comic-Con, Violet Ramis Steel, Tom Waltz, IDW Comics, diamond select, void vr, the void, diamond select toys, action figures, ghostbusters daughter, podcast book club, Dan Schoening, sdcc, egon spengler
#103 - "Group Photo" - July 20, 2015
In this week's episode, we'll talk about our spoiler policy moving forward, all of the awesome merchandise revealed at San Diego Comic-Con 2015 from IDW Publishing, Diamond Select Toys, Hollywood Collectables, and a curious Mattel display that has us wondering on their future with the franchise. Troy and Chris also talk about the group photo of Paul Feig's Ghostbusters (2016) cast that was officially released and speculate on what roles each of the actors might be playing in the reimagined series. Oh, and a special thanks to our very first sponsors: Challengers Comics in Chicago and Kerberos Productions!
Tagged: idw comics, san diego comic-con, hollywood collectables, mattel, diamond select toys, ghostbusters (2016), casting, podcasts, paul feig


Price includes a one color imprint on these custom economical book bags. Designed for thinner books and publications, these non-woven bags are constructed of heat sealed 80 GSM non-woven polypropylene. Features a die-cut handle and 2 1/2" gusset.
14.75" x 12.125" x 2.5"


More details have emerged about the contractual clause which has prevented Sheffield United from releasing Dean Hammond.
The midfielder, signed on loan from Leicester City last season, was placed on the transfer list by new manager Chris Wilder last week after triggering an option entitling him to a further 12 months at Bramall Lane.
Hammond, signed by Wilder's predecessor Nigel Adkins, is understood to have informed United's board of directors that he would be willing to accept a pay cut before the League One club published its 'retained and released' list on Friday evening.
Although it remains unclear whether this offer was accepted, The Star can confirm Wilder was powerless to off-load the 33-year-old because the temporary agreement Hammond negotiated in October was loaded in his favour.
The terms of that deal were designed to protect United's interests had Hammond proved a success during his spell in South Yorkshire.
But, following a series of disappointing displays, they also explain why co-owners Kevin McCabe and HRH Prince Abdullah Bin Mosaad Abdulaziz Al Saud, who heavily subsidised Adkins' squad last term, recently overhauled United's off-the-pitch operations.
McCabe pointedly referred to the fact an "owner representative" will now sit on the technical board responsible for "selecting and recruiting" players after parting company with Adkins less than a fortnight ago.
Wilder is set to embark upon a major rebuilding exercise this summer after releasing 10 players and telling another seven, including Hammond, they are free to look for other clubs, before the 2016/17 campaign gets underway.