text:
Q: android.os.NetworkOnMainThreadException for webservice (ksoap) I am new to android programming and trying to use webservice in this sample program:
I use Android 4.1 and my IDE is Eclipse Juno. I think the programming part is ok, but may be there is a problem about connecting.
package com.example.webserviceexample;

import java.io.IOException;

import org.ksoap2.SoapEnvelope;
import org.ksoap2.SoapFault;
import org.ksoap2.serialization.SoapObject;
import org.ksoap2.serialization.SoapPrimitive;
import org.ksoap2.serialization.SoapSerializationEnvelope;
import org.ksoap2.transport.HttpTransportSE;
import org.xmlpull.v1.XmlPullParserException;

import android.app.Activity;
import android.os.Bundle;
import android.view.Menu;
import android.view.View;
import android.view.View.OnClickListener;
import android.widget.Button;
import android.widget.EditText;
import android.widget.TextView;

public class MainActivity extends Activity {

    final static String NAMESPACE = "http://tempuri.org/";
    final static String METHOD_NAME = "CelsiusToFahrenheit";
    final static String SOAP_ACTION = "http://tempuri.org/CelsiusToFahrenheit";
    final static String URL = "http://www.w3schools.com/webservices/tempconvert.asmx";

    TextView sonuc;
    EditText deger;
    Button hesapla;

    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        deger = (EditText) findViewById(R.id.deger);
        sonuc = (TextView) findViewById(R.id.flag);
        hesapla = (Button) findViewById(R.id.hesapla);

        hesapla.setOnClickListener(new OnClickListener() {

            public void onClick(View v) {

                //request info
                SoapObject Request = new SoapObject(NAMESPACE, METHOD_NAME);
                Request.addProperty("Celcius",deger.getText().toString());

                //envelope
                SoapSerializationEnvelope soapEnvelope = new SoapSerializationEnvelope(SoapEnvelope.VER11);
                soapEnvelope.dotNet = true; //.NET = true, php = false

                //putting request to the envelope
                soapEnvelope.setOutputSoapObject(Request);

                //transferring data
                HttpTransportSE aht = new HttpTransportSE(URL); //prepare
                //start
                try {
                        aht.call(SOAP_ACTION, soapEnvelope);
                }
                catch (IOException e) {
                    e.printStackTrace();
                }
                catch (XmlPullParserException e)
                {
                    e.printStackTrace();
                }

                //waiting and getting response.
                String result;

                try {
                    // we are creating SoapPrimitive Object as waiting for simple variable.
                    result = "Fahrenheit:" + soapEnvelope.getResponse();

                    //writing the result to the textView
                    sonuc.setText(result);
                }
                catch (SoapFault e) {
                    e.printStackTrace();
                }

            }
        });
    }

    @Override
    public boolean onCreateOptionsMenu(Menu menu) {
        getMenuInflater().inflate(R.menu.activity_main, menu);
        return true;
    }
}

However, I get this error:
08-15 11:45:26.294: E/AndroidRuntime(641): FATAL EXCEPTION: main
08-15 11:45:26.294: E/AndroidRuntime(641): android.os.NetworkOnMainThreadException
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.os.StrictMode$AndroidBlockGuardPolicy.onNetwork(StrictMode.java:1117)
08-15 11:45:26.294: E/AndroidRuntime(641):  at java.net.InetAddress.lookupHostByName(InetAddress.java:385)
08-15 11:45:26.294: E/AndroidRuntime(641):  at java.net.InetAddress.getAllByNameImpl(InetAddress.java:236)
08-15 11:45:26.294: E/AndroidRuntime(641):  at java.net.InetAddress.getAllByName(InetAddress.java:214)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpConnection.<init>(HttpConnection.java:70)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpConnection.<init>(HttpConnection.java:50)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpConnection$Address.connect(HttpConnection.java:341)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpConnectionPool.get(HttpConnectionPool.java:87)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpConnection.connect(HttpConnection.java:128)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpEngine.openSocketConnection(HttpEngine.java:315)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpEngine.connect(HttpEngine.java:310)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpEngine.sendSocketRequest(HttpEngine.java:289)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpEngine.sendRequest(HttpEngine.java:239)
08-15 11:45:26.294: E/AndroidRuntime(641):  at libcore.net.http.HttpURLConnectionImpl.connect(HttpURLConnectionImpl.java:80)
08-15 11:45:26.294: E/AndroidRuntime(641):  at org.ksoap2.transport.ServiceConnectionSE.connect(ServiceConnectionSE.java:76)
08-15 11:45:26.294: E/AndroidRuntime(641):  at org.ksoap2.transport.HttpTransportSE.call(HttpTransportSE.java:146)
08-15 11:45:26.294: E/AndroidRuntime(641):  at org.ksoap2.transport.HttpTransportSE.call(HttpTransportSE.java:95)
08-15 11:45:26.294: E/AndroidRuntime(641):  at com.example.webserviceexample.MainActivity$1.onClick(MainActivity.java:61)
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.view.View.performClick(View.java:4084)
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.view.View$PerformClick.run(View.java:16966)
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.os.Handler.handleCallback(Handler.java:615)
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.os.Handler.dispatchMessage(Handler.java:92)
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.os.Looper.loop(Looper.java:137)
08-15 11:45:26.294: E/AndroidRuntime(641):  at android.app.ActivityThread.main(ActivityThread.java:4745)
08-15 11:45:26.294: E/AndroidRuntime(641):  at java.lang.reflect.Method.invokeNative(Native Method)
08-15 11:45:26.294: E/AndroidRuntime(641):  at java.lang.reflect.Method.invoke(Method.java:511)
08-15 11:45:26.294: E/AndroidRuntime(641):  at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:786)
08-15 11:45:26.294: E/AndroidRuntime(641):  at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:553)
08-15 11:45:26.294: E/AndroidRuntime(641):  at dalvik.system.NativeStart.main(Native Method)

So what can be the problem?

A: Try adding strictmode to your application.  In your onCreate method, add
    StrictMode.ThreadPolicy policy = new StrictMode.ThreadPolicy.Builder().permitAll().build();
    StrictMode.setThreadPolicy(policy);

A: You can't do Network operations on the main thread.
Checkout : http://developer.android.com/reference/android/os/AsyncTask.html
for painless background threading :)
EDIT: Since I am still getting up-votes for this answer even though it's several years old I would no longer suggest using AsyncTask. It has many known problems which are described here (http://blog.danlew.net/2014/06/21/the-hidden-pitfalls-of-asynctask/). Instead I would urge you to use Retrofit or one of the other nice http clients which handles the threading for you.

A: Full example:
import org.ksoap2.SoapEnvelope;
import org.ksoap2.serialization.SoapObject;
import org.ksoap2.serialization.SoapPrimitive;
import org.ksoap2.serialization.SoapSerializationEnvelope;
import org.ksoap2.transport.HttpTransportSE;

import android.app.Activity;
import android.os.AsyncTask;
import android.os.Bundle;
import android.util.Log;

public class MainActivity extends Activity {
    private String TAG ="Vik";

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        AsyncCallWS task = new AsyncCallWS();
        task.execute();
    }

    private class AsyncCallWS extends AsyncTask<Void, Void, Void> {
        @Override
        protected Void doInBackground(Void... params) {
            Log.i(TAG, "doInBackground");
            calculate();
            return null;
        }

        @Override
        protected void onPostExecute(Void result) {
            Log.i(TAG, "onPostExecute");
        }

        @Override
        protected void onPreExecute() {
            Log.i(TAG, "onPreExecute");
        }

        @Override
        protected void onProgressUpdate(Void... values) {
            Log.i(TAG, "onProgressUpdate");
        }

    }

    public void calculate()
    {
        String SOAP_ACTION = "http://tempuri.org/CelsiusToFahrenheit";
        String METHOD_NAME = "CelsiusToFahrenheit";
        String NAMESPACE = "http://tempuri.org/";
        String URL = "http://www.w3schools.com/webservices/tempconvert.asmx";

        try {
            SoapObject Request = new SoapObject(NAMESPACE, METHOD_NAME);
            Request.addProperty("Celsius", "32");

            SoapSerializationEnvelope soapEnvelope = new SoapSerializationEnvelope(SoapEnvelope.VER11);
            soapEnvelope.dotNet = true;
            soapEnvelope.setOutputSoapObject(Request);

            HttpTransportSE transport= new HttpTransportSE(URL);

            transport.call(SOAP_ACTION, soapEnvelope);
            SoapPrimitive resultString = (SoapPrimitive)soapEnvelope.getResponse();

            Log.i(TAG, "Result Celsius: " + resultString);
        }
        catch(Exception ex) {
            Log.e(TAG, "Error: " + ex.getMessage());
        }

        SOAP_ACTION = "http://tempuri.org/FahrenheitToCelsius";
        METHOD_NAME = "FahrenheitToCelsius";
        try {
            SoapObject Request = new SoapObject(NAMESPACE, METHOD_NAME);
            Request.addProperty("Fahrenheit", "100");

            SoapSerializationEnvelope soapEnvelope = new SoapSerializationEnvelope(SoapEnvelope.VER11);
            soapEnvelope.dotNet = true;
            soapEnvelope.setOutputSoapObject(Request);

            HttpTransportSE transport= new HttpTransportSE(URL);

            transport.call(SOAP_ACTION, soapEnvelope);
            SoapPrimitive resultString = (SoapPrimitive)soapEnvelope.getResponse();

            Log.i(TAG, "Result Fahrenheit: " + resultString);
        }
        catch(Exception ex) {
            Log.e(TAG, "Error: " + ex.getMessage());
        }
    }

}


text:
Q: Is there a way to convert this HQL to Criteria? This is the HQL I'm trying to convert:
UserRequest.executeQuery("Select ur.request from UserRequest ur
  where ur.user = :user and ur.request.type is null
  order by ur.request.dateCreated DESC",
[user: user],
[max: params.max?: 10, offset: params.offset?: 0])

I tried converting it to criteria with the code below but it gives me an error that No property request.type found
def list = UserRequest.createCriteria().list(max: params.max?: 10, offset: params.offset?: 0) {
     eq("user", user)
     isNull(request.type)
     order("request.dateCreated", "DESC")
}

Motivation for converting
In addition to having to write a separate query to get the total (for pagination) when using HQL the other reason I want to convert it to criteria is because I'll need to filter the results of this query based on parameters. For example:
def list = UserRequest.createCriteria().list(max: params.max?: 10, offset: params.offset?: 0) {
     eq("user", user)
     isNull("request.type")

     if (params.filtertype == "color")
       ilike("request.color", "%${params.filtervalue}%")
     order("request.dateCreated", "DESC")
}

If I try to do the above with HQL, it becomes way more cumbersome.
Update
Here is my class structure
class User {
    transient springSecurityService
    String username
    String password
}

Class Request {
    String type
}

class UserRequest implements Serializable{
    User user
    Request request

    static mapping = {
        id composite: ['user', 'request']
        version false
    }

    //equals & hashcode

    static UserRequest create (User user, Request request, boolean flush = true) {
        def d = new UserRequest (user: user, request: request)
                d.save(flush: flush, insert: true)
    }
}

A: Since your criteria query involves associations, try the following:
def list = UserRequest.createCriteria().list(max: params.max?: 10, offset: params.offset?: 0) {
     eq("user", user)
     request {
         isNull("type")
         order("dateCreated", "DESC")
     }
}

There's also a syntax variation that uses createAlias().  You might find that easier to work with, depending on the final structure of your query.  For more examples, see the "Querying Associations" section here: http://grails.org/doc/latest/guide/GORM.html#criteria

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}\label{sec0}

Being the central mathematical model in hydrodynamics, the Navier-Stokes and Euler equations permanently remain in the focus of both the analysis of PDEs and the theory of infinite dimensional dynamical systems and their attractors, see \cite{B-V,Ch-V-book,Fef06,FMRT,F95,Lad,L34,tao,T,T95} and the references therein for more details. Most studied is the 2D case where reasonable results on the global well-posedness and regularity of solutions as well as the results on the  existence of global attractors and their dimension are available. However, the global well-posedness  in
 the 3D case remains a mystery and even listed by the Clay institute of mathematics as one of the Millennium problems. This mystery inspires a comprehensive study of various modifications/regularizations of the initial Navier-Stokes/Euler equations (such as Leray-$\alpha$ model, hyperviscous Navier-Stokes equations, regularizations via $p$-Laplacian, etc.), many of which have a strong physical background and are of independent interest, see e.g. \cite{Camassa,HLT10,L34,Lopes,OT07} and the references therein.
\par
In the present paper we shall be dealing
 with the following regularized damped Euler system:
\begin{equation}\label{DEalpha}
\left\{
  \begin{array}{ll}
    \partial_t u+(\bar u,\nabla_x)\bar u+\gamma u+\nabla_x p=g,\  \  \\
    \operatorname{div} \bar u=0,\quad u(0)=u_0.
  \end{array}
\right.
\end{equation}
with  forcing $g$ and  Ekman damping term $\gamma u$,
$\gamma>0$. The damping term $\gamma  u$ makes the system dissipative and
is important in  various geophysical models~\cite{Ped}. Here and below  $\bar u$ is a smoothed (filtered)
vector field related with the initial velocity field $u$ as the
solution of the Stokes problem
\begin{equation}\label{0.bar}
u=\bar u-\alpha\Dx \bar u+\Nx q,\ \ \divv\bar u=0,
\end{equation}
where $\alpha>0$ is a given small parameter. In other words,
$$
\bar u=(1-\alpha A)^{-1}u,
$$
where $A:=\Pi\Dx$ is the Stokes operator and $\Pi$ is the Helmholtz--Leray projection to divergent free vector fields in the corresponding domain.
\par
System \eqref{DEalpha}, \eqref{0.bar} (at least in the conservative case $\gamma=0$) is
often referred to as the  simplified Bardina subgrid
scale model of turbulence, see \cite{BFR80,Bardina,Lay} for the derivation
of the model and further discussion, so in this paper
we  shall be calling  \eqref{DEalpha}  the damped Euler--Bardina equations.
We also mention that rewriting \eqref{DEalpha} in terms of the variable $\bar u$ gives
\begin{equation}\label{0.KV}
\Dt\bar u-\alpha\Dt\Dx\bar u+(\bar u,\Nx)\bar u+\gamma\bar u+\Nx p=\alpha\gamma\Dx \bar u+g
\end{equation}
which is a damped version of the so-called Navier--Stokes--Voight equations arising
in the theory of viscoelastic fluids, see \cite{Titi-Varga,Osk} for the details.
\par
Our main interest in the present paper is to study the dimension of
global attractors for system \eqref{DEalpha} in 2D and 3D paying main
attention to the most complicated 3D case. Note that, unlike the classical
Euler equations, Bardina-Euler equations can be interpreted as an ODE with bounded nonlineariry
in the
proper Hilbert space, so no problems with well-posedness arise, see \cite{Bardina}
and also section \S\ref{sec2} below, so the main aim of our study is
to get as sharp as possible bounds for the corresponding global attractors.
Each case $d=2$ and $d=3$ in
turn is studied in three different settings as  far as the boundary
conditions are concerned. More precisely, the system is studied
\begin{enumerate}
  \item on the torus $\Omega=\mathbb{T}^d=[0,2\pi]^d$. In this case the
      standard zero mean condition is imposed on  $u$, $\bar u$ and $g$;
  \item in the whole space $\Omega=\mathbb{R}^d$;
  \item in a bounded domain $\Omega\subset\mathbb{R}^d$  with Dirichlet boundary conditions
  for $\bar u$.
\end{enumerate}
 We denote by $W^{s,p}(\Omega)$ the standard Sobolev  space of distributions whose
 derivatives up to order $s$ belong to the Lebesgue space $L^p(\Omega)$. In the
 Hilbert case $p=2$ we will write $H^s(\Omega)$ instead of $W^{s,2}(\Omega)$.
 In order to work with velocity vector fields, we denote by ${\bf H}^s={\bf H}^s(\Omega)$
 the subspace of $[H^s(\Omega)]^d$ consisting of   divergence free vector fields.
 In the case of $\Omega\subset\R^d$ we assume in addition that vector fields from
 ${\bf H}^s$ satisfy Dirichlet boundary conditions and in the case of periodic
 boundary conditions $\Omega=\Bbb T^d$ we assume that these vector fields have zero mean.
 We also recall that equation \eqref{DEalpha} possesses the standard energy identity
 $$
 \frac12\frac d{dt}\(\|\bar u\|^2_{L^2(\Omega)}+\alpha\|\Nx \bar u\|^2_{L^2(\Omega)}\)+
 \gamma\(\|\bar u\|^2_{L^2(\Omega)}+\alpha\|\Nx \bar u\|^2_{L^2(\Omega)}\)=
(g,\bar u),
$$
where $(u,v)$ is the standard inner product in $[L^2(\Omega)]^d$.
For this reason it is natural to consider problem \eqref{DEalpha}
in the phase space ${\bf H}^1$  with  norm
$$
\|\bar u\|^2_{\alpha}:=\|\bar u\|^2_{L^2}+\alpha\|\Nx \bar u\|^2_{L^2}.
$$
Our first main result is the following theorem which gives an explicit upper
bound for the fractal dimension of the attractor in the 3D case.

\begin{theorem}Let $d=3$, let $\Omega$ be as described above, and
let  $g\in [L^2(\Omega)]^3$ (in the periodic case we assume also
that $g$ has zero mean). Then the solution semigroup $S(t)$ associated
with equation \eqref{DEalpha}  possesses
a global attractor $\mathscr A \Subset{\bf H}^1$ with finite fractal dimension
satisfying the following inequality:
\begin{equation}\label{0.est1}
\dim_F\mathscr A\le\frac{1}{12\pi}\frac{\| g\|_{L^2}^2}{\alpha^{5/2}\gamma^4}\,.
\end{equation}
The analogue of this estimate for the  2D case reads
\begin{equation}\label{0.est2}
\dim_F\mathscr A\le \frac{1}{16\pi}\frac{\| g\|_{L^2}^2}{\alpha^{2}\gamma^4}
\end{equation}
with the following improvement for the case when $\Omega=\Bbb T^2$ or $\Omega=\R^2$:
\begin{equation}\label{0.est3}
\dim_F\mathscr A\le \frac{1}{8\pi}\frac{\| \rot g\|_{L^2}^2}{\alpha\gamma^4}
\end{equation}
due to estimates related with the vorticity equation.
\end{theorem}
Since the general case $\gamma>0$ is reduced to the particular one with  $\gamma=1$ by
scaling $t\to\gamma^{-1}t$, $u\to\gamma^{-2}u$, $g\to\gamma^{-2}g$,
the most interesting in estimates \eqref{0.est1}, \eqref{0.est2}
and \eqref{0.est3} is the dependence of the RHS on $\alpha$. For
the viscous case of equations \eqref{DEalpha}
$$
\Dt u+(\bar u,\Nx)\bar u+\Nx q=\nu\Dx u+g
$$
the following estimate is proved in \cite{Bardina}:
$$
\dim_F\mathscr A\le C\frac{\|g\|_{L^2}^{6/5}}{\nu^{12/5}\alpha^{18/5}}
$$
for the case $\Omega=\Bbb T^3$. We see that even in the case $\nu=1$ this
estimate gives essentially worse dependence on $\alpha$ than our estimate \eqref{0.est1}.
The upper bounds for 3D Navier-Stokes-Voight equation obtained
in~\cite{Titi-Varga} give even worse dependence on the parameter $\alpha$ (like $\alpha^{-6}$).
 Estimates \eqref{0.est2} and \eqref{0.est3} have been proved
for $\Omega=\Bbb T^2$ in a recent paper~\cite{IZLap70}. The sharpness of these
estimates in the limit as $\alpha\to0$ was also established
there for the case of the 2D Kolmogorov flows. However, to the best of our knowledge, no
lower bounds for the
dimension of the attractor of the Euler--Bardina equations in 3D are available
in the literature.
\par
Our second main result covers this gap. Namely, we consider the 3D Kolmogorov
 flows on the  torus $\Omega=\Bbb T^3$ for equations \eqref{DEalpha} generated by the family
 of the right-hand sides parameterized by an integer parameter $s\in\mathbb{N}$:
\begin{equation}\label{0.kol}
g=g_s=\begin{cases} g_1=\gamma^2\lambda(s)\sin(sx_3),\\
g_2=0,\\ g_3=0,
            \end{cases}
\end{equation}
where $s\sim\alpha^{-1/2}$ and $\lambda(s)$ is a specially chosen amplitude, see \S\ref{sec4}.
Then, performing an accurate instability analysis for the linearization of equation \eqref{DEalpha}
on the corresponding Kolmogorov flow (in the spirit of \cite{Liu2}, see also \cite{IT1,IMT,Liu}),
we get the following result.

\begin{theorem} Let $\Omega=\Bbb T^3$ and let $\gamma>0$, and $\alpha>0$. Then in the limit
$\alpha\to0$ the integer parameter $s$ and  the amplitude $\lambda(s)$
can be chosen so that the corresponding forcing  $g=g_s$ of the form  \eqref{0.kol} produces
the global attractor $\mathscr A=\mathscr A_s$, whose
dimension satisfies the following lower bound:
\begin{equation}\label{0.est4}
\dim_F\mathscr A\ge c\frac{\| g\|_{L^2}^2}{\alpha^{5/2}\gamma^4},
\end{equation}
where $c>0$ is an absolute effectively computable constant.
\end{theorem}
Estimate \eqref{0.est4} shows that our upper bound \eqref{0.est1} is optimal. Again,
 to the best of our knowledge, this is the first optimal two-sided estimate for the attractor
 dimension in a 3D hydrodynamical problem.

In this connection we recall the celebrated upper bound in \cite{CFT} for the
attractor dimension of the
classical Navier--Stokes system on the 2D torus,
which is still  logarithmically larger than the corresponding lower bound in \cite{Liu}.
On the other hand, adding to the system an arbitrary fixed damping
makes it possible to obtain the estimate for the attractor dimension that is optimal
in the vanishing viscosity limit~\cite{IMT}.

\par
 We finally observe that the obtained lower estimates
for the attractor
dimension grow as $\alpha\to0$ in both 2D and 3D cases (and even are optimal for the case of tori), so
one may expect that the limit attractor $\Cal A_0$ (which corresponds to the case of non-modified damped Euler equation) is infinite dimensional. Indeed, the existence of the attractor $\Cal A_0$ in the proper phase space is well-known in 2D at least if $g\in W^{1,\infty}$, see \cite{CIZ} and references therein and we expect that some weaker version of the limit attractor $\Cal A_0$ can be also constructed in  3D using the trajectory approach, see \cite{Ch-V-book}, and the concept of dissipative solutions for 3D Euler introduced by P. Lions, see \cite{PLio}.
However, the situation with the dimension is much more delicate since the obtained lower bounds for the instability index on  Kolmogorov's flows are optimal for intermediate values of $\alpha$ only and do not provide any reasonable bounds for the limit case $\alpha=0$. Thus,
the question of finite or infinite-dimensionality of the limit attractor remains completely open even in the 2D case.

\par
The paper is organized as follows. The key estimates for the solutions of problem
\eqref{DEalpha} are derived in  \S\ref{sec2}. Global well-posedness and dissipativity
are also discussed there. The existence of a global attractor $\mathscr A$ is verified in
 \S\ref{sec3}. To make the proof independent of the choice of a (bounded or unbounded) domain
$\Omega$, we use the so called energy method for establishing the asymptotic compactness
of the associated semigroup.
\par

The upper bounds for its dimension are obtained in  \S\ref{sec4}
via the volume contraction method~\cite{B-V,CF85,T}. The essential role in getting
optimal bounds for the global Lyapunov exponents is played by the collective
Sobolev inequalities for $H^1$-orthonormal families
 proved in Appendix~\ref{sec5} based on the ideas of \cite{LiebJFA}.
 Their role is somewhat similar to the role of the Lieb--Thirring inequalities
 \cite{Lieb,LT} in the dimension estimates of the attractors
 of the classical Navier--Stokes equations \cite{B-V,T}.
The corresponding  inequality in the 2D case has also been  used in \cite{IZLap70}.
 Finally, the sharp lower bounds of the dimension for the case
 $\Omega=\Bbb T^3$ are obtained in  \S\ref{sec4} by adapting/extending the ideas
 of \cite{IZLap70,Liu2} to the 3D case.

\setcounter{equation}{1}
\section{A priori estimates, well-posedness and dissipativity }\label{sec2}

We start with the standard energy estimate, which looks the same in the 2D and  3D cases as
well as for the three types of boundary conditions.

\begin{proposition}\label{Prop:2.1} Let $u$ be a sufficiently regular solution of equation \eqref{DEalpha}.
Then the following dissipative energy estimate holds:
\begin{equation}\label{2.en}
\|\bar u(t)\|^2_\alpha\le \|\bar u(0)\|^2_\alpha e^{-\gamma t}+\frac1{\gamma^2}\|g\|^2_{L^2},
\end{equation}
where
\begin{equation}\label{2.norm}
\|\bar u\|^2_\alpha:=\|\bar u\|^2_{L^2}+\alpha\|\Nx\bar u\|^2_{L^2}.
\end{equation}
\end{proposition}
\begin{proof} Indeed, multiplying equation \eqref{DEalpha} by $\bar u$, integrating over $\Omega$ and using the relation between $u$ and $\bar u$ as well as the standard fact that the inertial term vanishes after the integration, we arrive at
\begin{multline}\label{2.en-est}
\frac d{dt}\left(\|\bar u\|^2_{L^2}+\alpha\|\nabla_x\bar u\|^2_{L^2}\right)+
2\gamma\left(\|\bar u\|^2_{L^2}+\alpha\|\nabla_x\bar u\|^2_{L^2}\right)=2(g,\bar u)\le\\\le
2\|g\|_{L^2}\|\bar u\|_{L^2}\le
\gamma\|\bar u\|^2_{L^2}+\frac1\gamma\|g\|_{L^2}^2.
\end{multline}
Applying the Gronwall inequality, we get the desired estimate \eqref{2.en}
and complete the proof.
\end{proof}
The next  corollary is crucial for our upper bounds for the attractor dimension.

\begin{corollary}\label{Cor2.en-av} Let $u$ be a sufficiently smooth solution of problem \eqref{DEalpha}. Then the following estimate holds:
\begin{equation}\label{2.est-int}
\limsup_{t\to\infty}\frac1t\int_0^t\|\Nx u(s)\|_{L^2}\,ds\le \frac1{\gamma\sqrt{2\alpha}}\|g\|_{L^2}.
\end{equation}
\end{corollary}
\begin{proof} Indeed, integrating estimate \eqref{2.en-est} over $t$, taking the limit $t\to\infty$ and using the fact that $\|u(t)\|_\alpha^2$ remains bounded (due to estimate \eqref{2.en}, we arrive at
$$
\limsup_{t\to\infty}\frac1t\int_0^t\|\Nx u(s)\|^2_{L^2}\,ds\le \frac1{2\alpha\gamma^2}\|g\|^2_{L^2}.
$$
Using after that the H\"older inequality
$$
\frac1t\int_0^t\|\Nx u(s)\|_{L^2}\,ds\le
 \(\frac1t\int_0^t\|\Nx u(s)\|^2_{L^2}\,dx\)^{1/2},
$$
we get the desired result and finish the proof of the corollary.
\end{proof}
We now turn to the two dimensional case without boundary. In this case, more accurate estimates are available due to the possibility to use the vorticity equation. Indeed,
applying $\operatorname{curl}$ to~\eqref{DEalpha} and
setting  $\omega=\operatorname{curl}u$, we obtain the vorticity equation
for $\omega$:
\begin{equation}\label{vort}
\partial_t\omega+(\bar u,\nabla_x)\bar\omega+\gamma\omega=\operatorname{curl} g,\ \omega=(1-\alpha\Delta_x)\bar\omega.
\end{equation}
 The estimates for the solution
on the torus $\mathbb{T}^2$ were derived in~\cite{IZLap70}. Although for
$\mathbb{R}^2$ they are formally the same, we reproduce them for the sake of completeness.

\begin{proposition} Let $u$ be a sufficiently smooth solution of \eqref{DEalpha},
where $\Omega=\mathbb{T}^2$ or $\mathbb{R}^2$ and let $\omega:=\rot u$ and $\bar\omega:=\rot \bar u$. Then, the following dissipative estimate holds:
\begin{equation}\label{2.en-vor}
\|\bar\omega(t)\|^2_{\alpha}\le \|\bar\omega(0)\|^2_{\alpha}e^{-\gamma t}+\frac1{\gamma^2}\|\rot g\|^2_{L^2}.
\end{equation}
\end{proposition}
\begin{proof}
Taking the scalar product of equation \eqref{vort} with $\bar\omega$, we see that the
nonlinear term vanishes and using that
\begin{equation}\label{omom}
(\omega,\bar\omega)=\|\bar \omega\|^2_{L^2}+\alpha\|\nabla_x\bar\omega\|^2_{L^2},
\end{equation}
 we obtain
\begin{multline}
\frac12\frac d{dt}\left(\|\bar \omega\|^2_{L^2}+\alpha\|\nabla_x\bar\omega\|^2_{L^2}\right)+
\gamma\left(\|\bar\omega\|^2_{L^2}+\alpha\|\nabla_x\bar\omega\|^2_{L^2}\right)=
(\rot g,\bar\omega)\le\\\le\|\rot g\|_{L^2}\|\bar\omega\|_{L^2}\le
\frac1{2\gamma}\|\rot g\|^2_{L^2}+\frac\gamma2\|\bar\omega\|^2_{L^2}.
\end{multline}
This gives the desired estimate \eqref{2.en-vor} by the  Gronwall inequality and
finishes the proof of the proposition.
\end{proof}
Analogously to Corollary \ref{Cor2.en-av}, we get the following estimate.

\begin{corollary}\label{Cor2.en-av2} Let $\Omega=\mathbb{T}^2$ or $\mathbb{R}^2$ and let
$u$ be a sufficiently smooth solution of problem \eqref{DEalpha}. Then the following estimate holds:
\begin{equation}
\limsup_{t\to\infty}\frac1t\int_0^1\|\Nx\bar u(s)\|_{L^2}\,ds\le \frac1\gamma\min\left\{\|\rot g\|_{L^2},\frac{\|g\|_{L^2}}{\sqrt{2\alpha}}\right\}.
\end{equation}
\end{corollary}
Indeed, the second inequality was already proved in Corollary
\ref{Cor2.en-av} and the first one is an immediate corollary of
\eqref{2.en-vor} and the fact that
$$
\|\nabla\bar u\|_{L^2}=\|\bar \omega\|_{L^2}.
$$

Let us conclude this section by discussing the well-posedness of
problem \eqref{DEalpha} and justification of the estimates obtained
above. We will consider below only the  3D case (the 2D case is
analogous and  even slightly simpler).
\par
We also note from the very beginning that equation \eqref{DEalpha}
can be rewritten in the form of   an ODE in a Hilbert space with
bounded nonlineariry. Indeed, applying the Helmholtz--Leray
projection $\Pi$ to both sides of \eqref{DEalpha} together with the
operator
$$
A_\alpha:=(1-\alpha A)^{-1},
$$
where $A=\Pi\Dx$ is the Stokes operator in $\Omega$, we arrive at
   \begin{equation}\label{1.ODE}
   \Dt\bar u+\gamma \bar u+B(\bar u,\bar u)=A_\alpha \Pi g, \ \bar u\big|_{t=0}=\bar u_0,
   \end{equation}

   where $B(\bar u,\bar v):=  A_{\alpha}\Pi\((\bar u,\Nx)\bar v\)$.
  \par
It is natural to  consider this system in the phase space
$\bar u\in{\bf H}^1(\Omega)$  with norm \eqref{2.norm}.
Then the nonlinear operator $B$ is bounded from $\mathbf{H}^1$
to $\mathbf{H}^{3/2}$:
\begin{equation}\label{1.B}
\|B(\bar u,\bar v)\|_{\mathbf{H}^{3/2}}\le
C_\alpha\|\bar u\|_{\alpha}\|\bar v\|_{\alpha},
\end{equation}
where $C_\alpha$ depends only on $\alpha$.
Indeed, if $\bar u,\bar v\in {\bf H}^1$, then by the Sobolev embedding theorem
 $\bar u,\bar v\in L^6(\Omega)$ and
 $(\bar u,\Nx)\bar v\in L^{3/2}(\Omega)$ by H\"older's inequality. Together with the
 $(L^{3/2}\to W^{2,3/2})$-boundedness of the operator $(1-\alpha A)^{-1}$,
 we get that $B(\bar u,\bar v)\in W^{2,3/2}(\Omega)$.
 Finally, the Sobolev embedding $W^{2,3/2}\subset H^{3/2}$ proves
 estimate \eqref{1.B}.

Thus, $B(\bar u,\bar u)$ is a regularizing
operator in ${\bf H}^1$ and equation \eqref{1.ODE} is an ODE in ${\bf H}^1$
with bounded nonlineariry. Therefore the local existence
and uniqueness of a solution as well as (an infinite) differentiability of the
corresponding local solution semigroup are straightforward corollaries of the
Banach contraction principle  or the
implicit function theorem, see e.g. \cite{henry} for the details.
Thus, to get the global well-posedness and dissipativity
 we only need to verify the proper a priori estimate.
Since this  has already been done in Proposition~\ref{Prop:2.1},
we have proved the following theorem.

    \begin{theorem}\label{Th1.exist} Let $\bar u_0\in {\bf H}^1(\Omega)$, $g\in [L^2(\Omega)]^d$
    (in the case of periodic BC we also assume that $g$ has zero mean).
     Then there exists a unique global solution $\bar u\in C([0,\infty),{\bf H}^1)$
    of problem \eqref{1.ODE} (which is simultaneously the unique solution of
    \eqref{DEalpha}). Moreover, the function
    $$
    t\to \|\bar u(t)\|_{L^2}^2+\alpha\|\Nx \bar u(t)\|^2_{L^2}
    $$
    is absolutely continuous
     and the following   energy identity holds:

    \begin{multline}\label{1.energy}
     \frac12\frac d{dt}\(\|\bar u(t)\|_{L^2}^2+\alpha\|\Nx \bar u(t)\|^2_{L^2}\)+\\+
     \gamma\(\|\bar u(t)\|_{L^2}^2+\alpha\|\Nx \bar u(t)\|^2_{L^2}\)=(g,\bar u).
    \end{multline}

    In particular, the dissipative estimate \eqref{2.en} holds for
    any solution $u$ of class $u\in C([0,\infty),{\bf H}^1)$.

    \end{theorem}

\begin{corollary}\label{Cor1.sem} Let the assumptions of Theorem \ref{Th1.exist} holds. Then equation \eqref{1.ODE}
generates a dissipative solution semigroup
\begin{equation}\label{1.s}
S(t)\bar u_0:= \bar u(t),\ \ t\ge0
\end{equation}
in the phase space ${\bf H}^1(\Omega)$. Moreover, $S(t)$ is $C^\infty$-differentiable for every fixed~$t$.
\end{corollary}
Indeed, the existence of the semigroup is an immediate corollary of the well-posedness proved in
the theorem and the differentiability follows from the ODE structure
of \eqref{1.ODE} and the fact that
the map $\bar u\to B(\bar u,\bar u)$ is $C^\infty$-smooth as
a map from ${\bf H}^1$ to ${\bf H}^1$.

\section{Asymptotic compactness and attractors}\label{s2}
In this section we construct a global attractor for the solution semigroup $S(t)$
generated  by problem \eqref{DEalpha}. We start with recalling the definition
of a weak and  strong   global attractor, see \cite{B-V,Ch-V-book} for more details.
We will mainly consider below the most complicated case $\Omega=\R^3$
since in the case of a bounded domain the asymptotic compactness
is an immediate corollary of the fact that $B(\bar u,\bar u)\in \mathbf{H}^{3/2}$
if $u\in {\mathbf H}^1$, see Remark \ref{Rem1.useless}.

\begin{definition} A set $\mathscr A_w\subset {\bf H}^1$ is a weak global attractor of the
semigroup $S(t)$ if
\par
1) $\mathscr A_w$ is a compact set in ${\bf H}^1$  with weak topology;
\par
2) $\mathscr A_w$ is strictly invariant, i.e., $S(t)\mathscr A_w=\mathscr A_w$;
\par
3) $\mathscr A_w$ attracts the images of all bounded sets in the weak topology of ${\bf H}^1$, i.e. for every bounded
set $B\subset {\bf H}^1$ and every neighbourhood $\mathcal O(\mathscr A_w)$ of the
attractor in the weak topology,
 there exists $T=T(\Cal O, B)$ such that
 $$
 S(t)B\subset\Cal O(\mathscr A_w) \ \text{for all} \ t\ge T.
 $$
Analogously, $\mathscr A_s$ is a strong attractor if it is compact in the strong
topology of ${\bf H}^1$, is strictly invariant and attracts the
images of bounded sets in the strong topology as well. Obviously
$$
\mathscr A_w=\mathscr A_s
$$
if both attractors exist.
\end{definition}

We will use the following criterion for verifying the existence of
an attractor, see \cite{B-V,T} for the details.

\begin{proposition}\label{Prop1.crit} Let the operators  operators $S(t)$ be
continuous in the weak topology for every fixed $t$ and let the semigroup $S(t)$
possess a bounded absorbing set $\Cal B$. The
latter means that for every bounded $B\subset {\bf H}^1$
there exists $T=T(B)$ such that
 $$
 S(t)B\subset \Cal B\ \ \text{for all}\ \ t\ge T.
 $$
 Then there exists a weak global attractor $\Cal A_w$ of the semigroup $S(t)$ which is
  generated by all complete (defined for all $t\in\R$) bounded solutions of problem \eqref{1.ODE}:

  \begin{equation}
\Cal A_w=\Cal K\big|_{t=0},
  \end{equation}

  where $\Cal K:=\{\bar u\in C_b(\R, {\bf H}^1),\ \ \bar u  \text{ solves } \eqref{1.ODE}\}$.
\par
Let, in addition, $S(t)$ be asymptotically compact on $\Cal B$. The latter means that
 for every sequence
 $\bar u_0^n\in \Cal B$ and every sequence $t_n\to\infty$, the sequence
 $$
 \{S(t_n)\bar u_0^n\}_{n=1}^\infty
 $$
 is precompact in the strong topology of ${\bf H}^1$. Then $\mathscr A_w$ is also a strong global attractor
 for the semigroup $S(t)$.
\end{proposition}
We start with verifying the existence of a weak attractor.

\begin{proposition}\label{Prop1.attr} Let the assumptions of Theorem \ref{Th1.exist} hold. Then the solution semigroup
 $S(t)$ generated by equation \eqref{DEalpha} possesses a weak global attractor $\Cal A_w$ in
  the phase space ${\bf H}^1$.
\end{proposition}
\begin{proof} The existence of a bounded absorbing set $\Cal B$ is an immediate corollary of the
 dissipative estimate \eqref{2.en}. We may take
 $$
 \Cal B:=\{\bar u\in {\bf H}^1,\ \ \|\bar u\|^2_{L^2}+\alpha\|\Nx\bar u\|^2_{L^2}\le
 \frac2{\gamma^2}\|g\|_{L^2}^2\}.
$$
Thus, we only need to check the weak continuity. Let $\bar u_0^n\in\Cal B$
be a sequence of  the initial data weakly converging to $\bar u_0$:
$\bar u_0^n\rightharpoondown \bar u_0$ in ${\bf H}^1$.
Denote by  $\bar u^n(t):=S(t)\bar u_0^n$
 the corresponding solutions. We need to check that for every fixed $T$,
 $\bar u^n(T)\rightharpoondown \bar u(T)$ in ${\bf H}^1$, where $\bar u(t):=S(T)\bar u_0$.
 \par

To see this  we recall that $\bar u^n$ is bounded uniformly with respect to
$n$ in $L^\infty(0,T;{\bf H}^1)$ due   to estimate \eqref{2.en}. Moreover,
from equation \eqref{1.ODE} we see also that $\Dt\bar u^n$ is
uniformly bounded in the same space. Thus, passing to a subsequence, if necessary,
 we may assume that
$\bar u^n(t)\rightharpoondown v(t)$ for every $t\in[0,T]$ and
$\Dt \bar u_n\rightharpoondown\Dt v$
in $L^2(0,T;{\bf H}^1)$ for some function $v(t)$ such that $v,\Dt v\in L^\infty(0,T;{\bf H}^1)$. So,
   it remains to verify that $v(t)=S(t)\bar u_0$ by passing to the limit in equations \eqref{1.ODE}
   for functions $\bar u^n$.
   \par
   This passing to the limit is obvious for linear terms, so we only need to prove the convergence
   of the nonlinear term $B(\bar u_n,\bar u_n)$. In turn, this is the same as to prove that, in the sense of distributions,
   $$
   (\bar u^n,\Nx)\bar u^n=\divv(\bar u^n\otimes \bar u^n)\rightharpoondown
    \divv(v\otimes v)=(v,\Nx)v.
   $$
The last statement will be proved if we check that
\begin{equation}\label{1.triv}
\bar u^n\otimes\bar u^n\rightharpoondown v\otimes v \ \ \text{in}\ \ L^2((0,T)\times\Omega).
\end{equation}
To verify \eqref{1.triv}, we recall that the sequence $\bar u^n\otimes\bar u^n$ is uniformly bounded in
$L^2$ due to  dissipative estimate \eqref{2.en} and the embedding
$H^1((0,T)\times\Omega)\subset L^4$.
Moreover, since  the embedding
$H^1((0,T)\times\R^3)\subset L^2((0,T);L^2_{loc}(\Omega))$ is compact,
we have the strong convergence $\bar u^n\to v$ in $L^2((0,T);L^2_{loc}(\Omega))$ and, therefore, the
 convergence $\bar u^n\to v$ almost everywhere. Since the sequence
 $\bar u^n\otimes\bar u^n$ is uniformly bounded in $L^2((0,T)\times\Omega)$,
 we may assume without loss of generality that
 it is weakly convergent to some  $\psi\in L^2((0,T)\times\Omega)$.
Along with the established convergence almost everywhere this implies
that $\psi=v\otimes v$, see e.g. \cite{lions}, and proves \eqref{1.triv}.
\par
Thus, we have proved that $v$ solves the equation \eqref{1.ODE} and by the
uniqueness $v(t)=\bar u(t)$. This finishes the proof of weak continuity of the
operators $S(t)$ and   the existence of a weak global attractor now follows
from Proposition \ref{Prop1.crit}. The theorem is proved.
\end{proof}

We are now ready to verify the existence of a strong global attractor.

\begin{proposition}\label{Prop1.attr-s} Let the assumptions of Theorem \ref{Th1.exist} hold. Then
 the solution semigroup $S(t)$ generated by equation \eqref{1.ODE} possesses a strong global attractor
  $\mathscr A=\mathscr  A_s$ in the phase space ${\bf H}^1$.
\end{proposition}

\begin{proof} According to Proposition \ref{Prop1.crit}, we only need to verify the asymptotic
compactness of $S(t)$ on $\Cal B$. We will use the so-called energy method for this
purpose, see \cite{ball,rosa} for more details.
\par
Let $\{\bar u^n_0\}\subset\Cal B$, let $t_n\to\infty$ be arbitrary and let
$\bar u^n(t):=S(t_n)\bar u^n_0$. Define also $\bar v^n(t):=\bar u^n(t+t_n)$. Then these
functions are defined on the time intervals $t\in[-t_n,\infty)$ and,
due to the existence
 of a weak global attractor, without loss of generality, we may assume that
  $\bar v(t)\rightharpoondown \bar u(t)$ in ${\bf H}^1$ for all $t\in\R$ to some
   complete trajectory $\bar u\in\Cal K$. In particular,

\begin{equation}\label{1.weak}
\bar v^n(0)=S(t_n)\bar u_0^n\rightharpoondown\bar u(0)
\end{equation}

and we only need to check that this convergence is strong.
\par
It is convenient to use the equivalent norm~\eqref{2.norm}  in the space ${\bf H}^1$.
Then, the strong convergence in \eqref{1.weak} will be proved if we verify that
\begin{equation}\label{norm-to-norm}
\|\bar v^n(0)\|^2_\alpha\to\|\bar u(0)\|^2_\alpha.
\end{equation}
To see this we integrate the energy identity \eqref{1.energy} for $\bar v^n(t)$
 in time and get
\begin{equation}\label{1.en-finite}
\|\bar v^n(0)\|^2_\alpha=\|\bar u_0^n\|^2_\alpha e^{-2\gamma t_n}+
\int_{-t_n}^0e^{2\gamma s}(g,\bar v^n(s))\,ds.
\end{equation}
Passing to the limit $n\to\infty$ in this relation and using the weak convergence of
$\bar v^n$ to  $\bar u$ and uniform boundedness of $\bar v^n$ and the initial data
$\bar u_0^n$,  we conclude that

 \begin{equation}\label{1.en-lim}
\lim_{n\to\infty}\|\bar v^n(0)\|^2_\alpha=\int_{-\infty}^0e^{2\gamma s}(g,\bar u(s))\,ds.
 \end{equation}

On the other hand, integrating the energy identity for the limit solution $\bar u$
in time, we arrive at
 \begin{equation}\label{1.en-lim1}
\|\bar u(0)\|^2_\alpha=\int_{-\infty}^0e^{2\gamma s}(g,\bar u(s))\,ds.
 \end{equation}

Equalities \eqref{1.en-lim} and \eqref{1.en-lim1} imply
\eqref{norm-to-norm}, therefore the convergence in \eqref{1.weak}
 is actually strong. Thus, the desired asymptotic compactness is proved and the proposition is also proved.
\end{proof}
\begin{remark}\label{Rem1.useless}
{\rm Since the operator $B(\bar u,\bar u)$ is
regularizing, one can easily
increase the regularity of the global attractor $\mathscr A$ using the  decomposition
of the  semigroup into the decaying linear part and the regularizing nonlinear part
(see \cite{Aro}):
 $$
 S(t):=L(t)+K(t),
 $$
 where $v(t)=L(t)\bar u_0 $ solves
 $$
 \Dt v+\gamma v=0,\ \ v\big|_{t=0}=\bar u_0
 $$
 and $w(t):=K(t)\bar u_0$ satisfies
 $$
\Dt w+\gamma w+B(\bar u,\bar u)= A_\alpha \Pi g,\ \ w\big|_{t=0}=0.
 $$
Combining this decomposition with bootstrapping arguments, we may check that the
regularity of the attractor $\mathscr A$ is restricted by the regularity of $g$ only and
it will be $C^\infty$-smooth if $g\in H^\infty(\R^3)$. Moreover, using the proper
weighted estimates, see \cite{MirZel}, we may
get the estimates on the rate of decay for solutions belonging to the attractor as
$|x|\to\infty$ in terms of the decay rate of $g$ which clarify the reason why
$\mathscr A$ is compact. However,
all these estimates do not seem  very helpful for estimation of the attractor
dimension     (since they grow rapidly with respect to $\gamma,\alpha\to0$) and
therefore  we will not go into  more      details here.}
\end{remark}

 \setcounter{equation}{0}

\section{Upper bounds for the fractal dimension}\label{sec3}
In this section we derive upper bounds for the fractal dimension of
the attractor~$\mathscr A$. As usual for the Navier--Stokes type
equations, these bounds will be obtained by means of the  volume
contraction method, see \cite{B-V,CF85,T} and the references
therein. On the analytical side, the Lieb--Thirring inequalities
for $L^2$-orthonormal families \cite{Lieb,LT} which are an
indispensable tool for the dimension estimates of the attractors
for the Navier--Stokes equations are replaced in our case by the
collective Sobolev inequalities for $H^1$-orthonormal families and
are proved in the Appendix \ref{sec5}.

Furthermore, since system \eqref{DEalpha} in the 2D
case   has already been  studied in \cite{IZLap70}
(for the case $\Omega=\Bbb T^2$), we will concentrate here on the 3D case only.

\begin{theorem}\label{Th:est} Suppose that  $\Omega$ is  either the  3D torus
$\Bbb T^3$, or  a bounded domain $\Omega\subset\R^3$
(endowed with Dirichlet BC), or the whole space $\Omega=\R^3$.
Let $g\in [L^2(\Omega)]^d$ (in the case of $\Bbb T^3$ we also assume that
$g$ has zero mean). Then the
global attractor $\mathscr A$ corresponding to the regularized
damped Euler system \eqref{DEalpha}
has  finite fractal dimension satisfying the following estimate:
\begin{equation}\label{dim-est}
\dim_F\mathscr{A}\le
\frac{1}{12\pi}\frac{\| g\|_{L^2}^2}{\alpha^{5/2}\gamma^4}\,.
\end{equation}
\end{theorem}

\begin{proof} The solution semigroup   $S(t): {\bf H}^1\to
{\bf H}^1$ is smooth with respect to the initial data (see Corollary \ref{Cor1.sem}),
 so we only need to estimate the $n$-traces for the linearization of equation
 \eqref{1.ODE} over trajectories on the attractor. This
 linearization of  \eqref{DEalpha} reads:

 \begin{equation}
 \begin{cases}
 \Dt \bar\theta=-\gamma\bar\theta-B(\bar u(t),\bar\theta)-B(\bar\theta,\bar u(t)) =:L_{u(t)}\bar \theta,\\
\operatorname{div} \bar\theta=0,\  \bar\theta\big|_{t=0}=\bar\theta_0\in \mathbf H^1(\Omega),
\end{cases}
\end{equation}
where  $B(\bar u,\bar v):=A_{\alpha}\Pi\((\bar u,\Nx)\bar v\)$. In order to
 utilize the  well-known cancelation property
$$
((\bar u,\Nx)\bar \theta,\bar\theta)\equiv 0
$$
for the inertial term in the Navier-Stokes equations, it is natural to endow the space ${\bf H}^1$ with the scalar product
\begin{equation}\label{scal-alpha}
(\bar\theta,\bar\xi)_\alpha=(\bar\theta,\bar\xi)+\alpha(\Nx\bar\theta,\Nx\bar\xi)=((1-\alpha A)\bar\theta,\bar\xi)
\end{equation}
associated with the norm \eqref{2.norm}. Then, using that $\Pi A_\alpha=A_\alpha$ and $\Pi\bar\theta=\bar\theta$, we get the cancelation
\begin{multline*}
(B(\bar u,\bar\theta),\bar\theta)_\alpha=
\left(A_\alpha \Pi(\bar u,\Nx)\bar\theta,(1-\alpha\Delta_x)\bar\theta\right)=\\=
\left(A_\alpha \Pi(\bar u,\Nx)\bar\theta,(1-\alpha\Pi\Delta_x)\bar\theta\right)=
\left(\Pi(\bar u,\Nx)\bar\theta,\bar\theta\right)=
((\bar u,\Nx)\bar\theta,\bar\theta)\equiv0
\end{multline*}
of the most singular term $B(\bar u,\bar\theta)$ and, therefore, only the more
regular term $B(\bar\theta,\bar u)$ will impact the trace estimates.
\par
Following the general strategy, see e.g. \cite{T}, the $n$-dimensional volume
 contraction factors $\omega_n(\mathscr A)$ (=the sums of the first $n$
global Lyapunov exponents) which control the dimension can be estimated from
above by the following numbers:
$$
q(n):=\limsup_{t\to\infty}\sup_{u(t)\in\mathscr A}\sup_{\{\bar\theta_j\}_{j=1}^n}\frac1t\int_0^t
\sum_{j=1}^n(L_{u(\tau)}\bar\theta_j,\bar\theta_j)_\alpha d\tau,
$$
where the first (inner) supremum is taken over all orthonormal families $\{\bar\theta_j\}_{j=1}^n$ with respect to the scalar product $(\cdot,\cdot)_\alpha$ in ${\bf H}^1$:
\begin{equation}\label{alpha-orth}
(\bar\theta_i,\bar\theta_j)_\alpha=\delta_{i\,j},\quad\operatorname{div} \theta_j=0,
\end{equation}
and the second (middle) supremum  is over all   trajectories $u(t)$ on the attractor
$\mathscr A$. Then,
using the cancellation mentioned above together with the pointwise estimate \eqref{dpointwise} proved in Appendix \ref{sB}, we get
\begin{multline}
\sum_{j=1}^n(L_{u(t)}\bar\theta_j,\bar\theta_j)_\alpha=
-\sum_{j=1}^n\gamma\|\bar\theta_j\|^2_{\alpha}-
\sum_{j=1}^n((\bar\theta_j,\Nx)\bar u,\bar\theta_j)\le\\\le
-\gamma n+\sqrt{\frac23}\int_\Omega\rho(x)|\nabla_x\bar u(t,x)|\,dx
\le -\gamma n+\sqrt{\frac23}\|\Nx \bar u(t)\|_{L^2}\|\rho\|_{L^2},
\end{multline}
 where
$$
\rho(x)=\sum_{j=1}^n|\bar\theta_j(x)|^2.
$$
We now use  estimate
 \eqref{R23alpha} from  Appendix \ref{sec5}
\begin{equation}\label{Lieb-for-us}
\|\rho\|_{L^2}\le\frac1{2\sqrt{\pi}}\frac{n^{1/2}}{{\alpha}^{3/4}}
\end{equation}
and obtain
\begin{equation}
\sum_{j=1}^n(L_{u(t)}\bar\theta_j,\bar\theta_j)_\alpha\le-\gamma n+\frac1{\sqrt 6\pi}\frac{n^{1/2}}{\alpha^{3/4}}\| \Nx\bar u(t)\|_{L^2}.
\end{equation}
Finally, using \eqref{2.est-int}, we arrive at
$$
q(n)\le-\gamma n +\frac{1}{2\sqrt{3\pi}}\frac{n^{1/2}}{{\alpha}^{5/4}}
\frac{\| g\|_{L^2}}{\gamma}\,.
$$
It only remains to recall that, according to the general theory,
$\omega_n(\mathscr A)\le q(n)$ and
any number $n^*$  for which $\omega_{n^*}(\mathscr A)\le0$  and
$\omega_{n}(\mathscr A)<0$ for $n>n^*$ is  an  upper bound both for the
Hausdorff \cite{B-V,T} and
the fractal \cite{Ch-I2001,Ch-I} dimension of the global attractor
$\mathscr A$. This gives the desired estimate
$$
\dim_F\mathscr{A}\le
\frac{1}{12\pi}\,\frac{\| g\|_{L^2}^2}{\alpha^{5/2}\gamma^4}
$$
and finishes the proof of the theorem.
\end{proof}
\begin{remark}\label{Rem:2d}
{\rm
Estimates~\eqref{0.est2} and \eqref{0.est3} for $\mathbb{T}^2$  (and the fact that it is sharp)  were proved in~\cite{IZLap70}.
The upper bound for  $\mathbb{R}^2$ is exactly the same once
we now know~\eqref{R23alpha} for $\mathbb{R}^2$. For a bounded domain
we only need to replace in the proof in~\cite{IZLap70} the estimates of the solutions on the attractor
by \eqref{2.est-int}. Alternatively, one can go through the
proof of Theorem~\ref{Th:est} and replace the 3D constants  by their
2D counterparts accordingly.
}
\end{remark}

 \setcounter{equation}{0}
\section{Sharp lower bound on $\mathbb{T}^3$ }\label{sec4}
The aim of this section is to show that estimate~\eqref{dim-est}
for system \eqref{DEalpha} on $\mathbb{T}^3=[0,2\pi]^3$ is
sharp in the limit as $\alpha\to0$.

We consider a family of right-hand sides
\begin{equation}\label{Kolmf}
g=g_s=\begin{cases}g_1=\gamma\lambda(s)\sin sx_3,\\
g_2=0,\\
g_3=0,\end{cases}
\end{equation}
depending only on $x_3$ and parameterized by $s\in \mathbb{N}$,
$s\gg1$. The amplitude function $\lambda(s)$  will be specified in the course
of the proof. Corresponding to the family $g_s$ is the family of
stationary solutions of \eqref{DEalpha}
\begin{equation}\label{u0}
\vec u_0(x_3)=\begin{cases}u_0(x_3)=\lambda(s)\sin sx_3,\\
0,\\
0,\end{cases}
\end{equation}
with $p=0$. In fact,
$$
\bar {\vec u}_0=(1-\alpha\Dx)^{-1}\vec u_0=(\bar u_0,0,0)^T
$$ also depends
only on $x_3$ and therefore $(\bar{\vec u}_0,\Nx)\bar{\vec u}_0=0$.

We now consider system~\eqref{DEalpha} linearized on the stationary
solution \eqref{u0}
\begin{equation}\label{linear}
\left\{
\aligned
    \partial_t w+\bar u_0\frac{\partial\bar w}{\partial x_1}
    +\bar w_3 \frac{\partial\bar u_0}{\partial x_3}e_1+\gamma w+\nabla_x q=0, \\
    \operatorname{div} w=0,
\endaligned
\right.
\end{equation}
where $e_1=(1,0,0)^T$ and  $\bar w=(1-\alpha\Dx )^{-1}w$. The
standing assumption~is
\begin{equation}\label{zero}
    \int_{\mathbb{T}^3}w(x,t)dx=0.
\end{equation}
We shall  look for the solution of the linear problem
\eqref{linear} in the form
\begin{equation}\label{exp}
 w(x,t)=\left(\aligned
 w_1(x_3)\\ w_2(x_3)\\ w_3(x_3)
 \endaligned
 \right)
 e^{i(ax_1+bx_2-act)},\qquad
 q(x,t)=q(x_3)e^{i(ax_1+bx_2-act)},
\end{equation}
where $a,b\in\mathbb{Z}$ so that $w$ and $q$ are $2\pi$-periodic in
each $x_i$. If such a solution of~\eqref{linear} is found, then
substituting~\eqref{exp} into~\eqref{linear} and setting $t=0$ we
see that
$$
w(x,0)=\left(\aligned
 w_1(x_3)\\ w_2(x_3)\\ w_3(x_3)
 \endaligned
 \right)
 e^{i(ax_1+bx_2)}
 $$
is a vector-valued eigenfunction of the stationary operator
\begin{equation}\label{L3}
L_3(\vec u_0)w=\bar u_0\frac{\partial\bar w}{\partial x_3}
    \bar u+\bar w_3 \frac{\partial\bar u_0}{\partial x_3}e_1+\gamma w+\nabla_x q
\end{equation}
and $iac$ is the corresponding eigenvalue. If $\mathrm{Re}(iac)<0$,
then the corresponding mode is unstable.

We substitute \eqref{exp} into \eqref{linear} and obtain the system
\begin{equation}\label{abc}
\left\{
  \begin{array}{ll}
 -\gamma w_1-ia(\bar u_0\bar w_1-c w_1)=iaq+\bar w_3u_0' \\
    -\gamma w_2-ia(\bar u_0\bar w_2-c w_2)=ibq,  \\
     -\gamma w_3-ia(\bar u_0\bar w_3-c w_3)=q',  \\
 ia w_1+ibw_2+w_3'=0,
  \end{array}
\right.
\end{equation}
where $\ '=\partial\,/\partial x_3$.
\begin{lemma}\label{L:a0}
There are no unstable solutions of equation \eqref{linear}
 $$
\partial_tw=L_3(\vec u_0)w
$$
that can be written in the form~\eqref{exp} with $a=0$.
\end{lemma}
\begin{proof}
Let $a=0$. Then the solutions of \eqref{linear} are sought in the
form
$$
 w(x,t)=\left(\aligned
 w_1(x_3)\\ w_2(x_3)\\ w_3(x_3)
 \endaligned
 \right)
 e^{i(bx_2-ct)},\qquad
 q(x)=q(x_3)e^{i(bx_2-ct)},
$$
and \eqref{abc} goes over to
$$
\left\{
  \begin{array}{ll}
 -\gamma w_1+ic w_1=\bar w_3u_0' \\
    -\gamma w_2+ic w_2=ibq,  \\
     -\gamma w_3+ic w_3=q',  \\
 ibw_2+w_3'=0.
  \end{array}
\right.
$$
Let $b\ne0$. Then $w_2=-w_3'/(ib)$. Substituting this into the
second  equation and differentiating the third with respect to
$x_3$ we obtain
$$
q''=b^2q,
$$
which gives that $q=0$, since $q$ is periodic. Since we are looking
for unstable solutions, it follows that $\mathrm{Re}(ic)<0$ and
therefore $-\gamma+ic\ne0$. This gives that $w_2=w_3=0$, and,
finally, $w_1=0$.

If $a=b=0$, then $w_3'=0$, and $w_3=0$ by periodicity and zero mean
condition. This gives $q=0$ and $w_1=w_2=0$. The proof is
complete.
\end{proof}

\subsection{Squire's transformation} We now reduce the 3D
instability analysis to the instability analysis of the transformed
2D problem. The key role is played by the Squire's transformation
(see \cite{Squire}, \cite {DrazinReid}, \cite{Liu2}).

Since we a looking for unstable solutions of \eqref{linear}, in
view of Lemma~\ref{L:a0} we may assume that $a\ne0$ in \eqref{abc}.
Multiplying the first equation in~\eqref{abc} by $a$ and the second
by~$b$ a adding up the results, we obtain
\begin{equation}\label{Squire}
\left\{
  \begin{array}{ll}
 -\widehat\gamma \widehat w_1-i\widehat a(\bar u_0\bar {\widehat{w}_1}-
\widehat c\, \widehat w_1)=i\widehat a\,  \widehat q+\bar{\widehat{w}}_3u_0', \\
-\widehat\gamma \widehat w_3-i\widehat a(\bar u_0\bar {\widehat{w}_3}-
\widehat c\, \widehat w_3)=\widehat q\,',  \\
 i\widehat a\, \widehat w_1+\widehat w'_3=0,
  \end{array}
\right.
\end{equation}
where
\begin{equation}\label{tilde}
\aligned
\widehat a\,^2=a^2+b^2,\qquad &\widehat w_1=\frac{aw_1+bw_2}{\widehat a},\qquad
\widehat w_3=w_3,\\
\widehat\gamma=\gamma\frac{\widehat a}a,\qquad
&\widehat q=q\frac{\widehat a}a,\qquad \widehat c=c.
\endaligned
\end{equation}

The solutions of this problem on the 2d torus
$$\mathbb{T}^2_{|\widehat a|}=x_1\in [0,2\pi/|\widehat a|\,],\
x_3\in[0,2\pi]$$
 are sought in the form
\begin{equation}\label{exp2}
 \widehat w(x_1,x_3,t)=\left(\aligned
 \widehat w_1(x_3)\\ \widehat w_3(x_3)
 \endaligned
 \right)
 e^{i(\widehat a\,x_1-\widehat a\widehat ct)},\quad
 \widehat q(x_1,x_3,t)=q(x_3)e^{i(\widehat a\,x_1-\widehat a\widehat ct)},
\end{equation}
and if such a solution is found, then the vector function
\begin{equation}\label{w-unstable}
\widehat w(x_1,x_3,0)=\left(\aligned
 \widehat w_1(x_3)\\ \widehat w_3(x_3)
 \endaligned
 \right)
 e^{i\widehat a\,x_1}
\end{equation}
is a vector-valued eigenfunction with
 eigenvalue  $i\widehat a\,\widehat c$ of the stationary operator
\begin{equation}\label{L2}
L_2(\vec u_0)\widehat w=\widehat \gamma \widehat w+\bar u_0
\frac{\partial\widehat{\bar w}}{\partial x_1}
    +\widehat{\bar w}_3 \frac{\partial\bar u_0}{\partial x_3}e_1+\Nx\widehat  q,
\quad \operatorname{div}\widehat w=0,
\end{equation}
on  $\mathbb{T}^2_{|\widehat a|}$, where the stationary solution
and the generating right-hand side are
\begin{equation}\label{u02}
\vec u_0(x_3)=\begin{cases}u_0(x_3)=\lambda(s)\sin sx_3,\\
0,\end{cases}\
g_s(x_3)=\begin{cases}g_1(x_3)=\lambda(s)\widehat\gamma\sin sx_3,\\
0,\end{cases}
\end{equation}
and where as before
$\bar u_0=\bar u_0(x_3)=(1-\Dx)^{-1}u_0.$

To avoid unnecessary complications we assume in what follows that
$$
\sqrt{a^2+b^2}=\widehat a>0,\quad\ a>0,
$$
and formulate the main result on the Squire's reduction of the 3D
instability analysis to the 2D case.

\begin{lemma}\label{L:Squire}
Let $\widehat w$ in \eqref{w-unstable} be an unstable eigenfunction of
the opera\-tor~\eqref{L2} on the torus
$\mathbb{T}^2_{\widehat a}=[0,2\pi/\widehat a\,]\times[0,2\pi]$. Then for
any pair of integers $a,b\in\mathbb{Z}$ with
$$
a^2+b^2=\widehat a^2
$$
there exist an unstable solution of system~\eqref{abc} on $\mathbb{T}^3=[0,2\pi]^3$.
\end{lemma}
\begin{proof}
Once the $\widehat\cdot$\,-\,variables are known, $q,w_3,c$ and
$\gamma$ are found from \eqref{tilde}. It remains to find $w_1$ and
$w_2$. We consider the second equation in~\eqref{abc}:
$$
{\rm A}w_2:=(-\gamma +iac)w_2-ia\bar u_0\bar w_2=ibq.
$$
Since $\widehat w$ is unstable, $\mathrm{Re}(iac)<0$ and therefore
$\mathrm{Re}(-\gamma+iac)<0$. Suppose that ${\rm A}w_2=0$ for some $w_2$. Taking
the scalar product in (complex) $L^2(0,2\pi)$
with $w_2$ and taking into account that the second term is purely imaginary
we obtain for the real part
$$
\mathrm{Re}(-\gamma+iac)\|w_2\|_{L^2}^2=0,
$$
which gives that $w_2=0$, and $A$ has a trivial kernel.
In addition, $A$ is a Fredholm operator,  since the second term is
compact (smoothing). Hence it has a bounded inverse,
and since $q$ is known, we have  found $w_2$. Finally,
$$w_1=(\widehat a\widehat w_1-bw_2)/a.$$
\end{proof}

\subsection{Instability analysis on $\mathbb{T}^2$}  We now have to recall the
instability analysis for the 2D problem that was carried out in detail in
our previous work~\cite{IZLap70}.
The problem was studied on the standard torus $\mathbb{T}^2=[0,2\pi]^2$
and we now denote the second coordinate by $x_3$, so that
$x_1$, $x_3$ are the coordinates on $\mathbb{T}^2$. The family of the forcing terms
and the corresponding stationary solutions are as in~\eqref{u02}
and the linearized stationary operator is precisely~\eqref{L2}.
Applying $\operatorname{curl}$ to~\eqref{L2}
 we obtain the equivalent  scalar
operator in terms of the vorticity  whose spectrum was  studied in \cite{IZLap70}
\begin{equation}\label{linvort}
 \aligned{\mathcal L}_{s}\omega:=
    &J\bigl((\Dx-\alpha\Dx^2)^{-1}\omega_s,(1-\alpha\Dx)^{-1}\omega\bigr)+\\+
&J\bigl((\Dx-\alpha\Dx^2)^{-1}\omega,(1-\alpha\Dx)^{-1}\omega_s\bigr)+
\gamma\omega=
-\sigma\omega,
\endaligned
\end{equation}
where
$$
J(a,b)=\nabla a\cdot\nabla^\perp b=\partial_{x_1}a\,\partial_{x_3}b-
\partial_{x_3} a\,\partial_{x_1} b,
$$
and
$$
\omega_s=\operatorname{curl}\vec u_0=-\lambda(s)s\cos sx_2,\qquad
\omega=\operatorname{curl}\widehat w.
$$
The following result was proved in \cite{IZLap70} (see Theorem~4.1 and Corollary~4.2.)
\begin{theorem}\label{Th:2d}
Given a large  integer $s>0$
let a fixed pair of integers $t,r$ belong to a bounded region
$A(\delta)$ defined by conditions
\begin{equation}\label{cond}
t^2+r^2<s^2/3,\quad t^2+(-s+r)^2>s^2,\quad t^2+(s+r)^2>s^2, \quad t\ge\delta s,
\end{equation}
where $0<\delta<1/\sqrt{3}$.
There exists an absolute constant $c_1$ such that
for
\begin{equation}\label{lambda}
\lambda\ge\lambda_2(s,\gamma)=c_1 \gamma\frac{(1+\alpha s^2)^2}s
\end{equation}
in~\eqref{u02} the linear operator ${\mathcal L}_{s}$ on the torus
$\mathbb{T}^2=[0,2\pi]^2$ has a real
negative (unstable) eigenvalue $\sigma<0$ of multiplicity $2$.
The corresponding eigenfunctions are
\begin{equation}\label{omega}
\aligned
&\omega^1(x_1,x_3)=\sum_{n=-\infty}^\infty a_{t,sn+r}\cos(tx_1+(sn+r)x_3),\\
&\omega^2(x_1,x_3)=\sum_{n=-\infty}^\infty a_{t,sn+r}\sin(tx_1+(sn+r)x_3).
\endaligned
\end{equation}
\end{theorem}

We now  observe that $\omega^1$ and $\omega^2$ in~\eqref{omega} are the real and imaginary parts of the
complex-valued eigenfunction
$$
\omega^1(x_1,x_3)+i\omega^2(x_1,x_3)=
\left[\sum_{n=-\infty}^\infty a_{t,sn+r}e^{i(sn+r)x_3}\right]e^{itx_1}.
$$

Recovering the corresponding divergence free vector function, that is, applying the
operator $\Nx^\perp\Dx^{-1}$, we obtain an unstable vector valued eigenfunction
of the operator $L_2(\vec u_0)$ written in the required form \eqref{w-unstable}:
$$
 w(x_1,x_3)=\left(\aligned
  w_1(x_3)\\ w_3(x_3)
 \endaligned
 \right)
 e^{it\,x_1}.
$$

For the 3D instability analysis below we need to repeat the
construction of an unstable eigenmode on the torus
$\mathbb{T}^2_\varepsilon$ with $x_1\in[0,2\pi/\varepsilon],x_2\in[0,2\pi]$,
where $\varepsilon>0$ is arbitrary (not necessarily small).
\begin{proposition}\label{Prop:eps}
Let $r$ and $t':=t\varepsilon$ belong to the region $A(\delta)$:
\begin{equation}\label{cond1}
{t'}^2+r^2<s^2/3,\quad {t'}^2+(-s+r)^2>s^2,\quad {t'}^2+(s+r)^2>s^2,
\quad {t'}\ge\delta s.
\end{equation}
Let $\lambda$ be defined in \eqref{lambda} and let $g_s$ and $\vec u_0$ be
the same as before but in two dimensions:
$$
g_s(x_3)=(\gamma\lambda(s)\sin sx_3,0)^T,\quad
\vec u_0(x_3)=(\lambda(s)\sin sx_3,0)^T.
$$
Then there exists an unstable solution
\begin{equation}\label{veceigmode}
 w(x_1,x_3)=\left(\aligned
  w_1(x_3)\\ w_3(x_3)
 \endaligned
 \right)
 e^{it\varepsilon\,x_1},\qquad x\in \mathbb{T}^2_\varepsilon.
\end{equation}
of the form \eqref{w-unstable} of the operator~\eqref{L2}
on the torus $ \mathbb{T}^2_\varepsilon.$
\end{proposition}
\begin{proof}
Following the proof of Theorem~4.1 in \cite{IZLap70} we see that a word for word
repetition of it shows that if  $t'=\varepsilon t ,r$
satisfy~\eqref{cond1}, then the corresponding operator \eqref{linvort}
has an unstable (real negative) eigenvalue of multiplicity two
with eigenfunctions
$$
\aligned
&\omega^1(x_1,x_3)=\sum_{n=-\infty}^\infty a_{t,sn+r}\cos(t\varepsilon x_1+(sn+r)x_3),\\
&\omega^2(x_1,x_3)=\sum_{n=-\infty}^\infty a_{t,sn+r}\sin(t\varepsilon x_1+(sn+r)x_3),
\endaligned
$$
from which we construct the required vector valued complex eigenfunction
\eqref{veceigmode} as before.
\end{proof}

It is convenient for us to single out a small rectangle $D$ in
the $(t',r)$-plane  inside the
region $A(\delta)$ defined by~\eqref{cond1}, see Fig.\ref{fig1}:
\begin{equation}\label{rectan}
|r|\le c_2s,\quad 0<c_3s\le t'=t\varepsilon\le c_4s.
\end{equation}
Here $\delta=\delta^*\in(0,1/\sqrt{3})$ is fixed, and all the constants $c_i$ are absolute constants,
 whose explicit values can easily be specified.

\begin{figure}[htb]
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  \providecommand\rotatebox[2]{#2
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    \setlength{\unitlength}{419.00001526bp
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  \els
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  \f
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  \makeatothe
  \begin{picture}(1,0.75178995
    \put(0,0){\includegraphics[width=\unitlength]{Adelta4.eps}
    \put(0.369,0.62){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\delta s$}}
    \put(0.375,0.68){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$r$}}
    \put(0.65,0.36){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$t',t$}}
    \put(0.49,0.415){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$A(\delta)$}}
    \put(0.49,0.38){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$D$}}
  \end{picture
\endgrou
\caption{The region $A(\delta)$ and the rectangle $D$.}
\label{fig1}
\end{figure}

\subsection{3D lower bound} We can now formulate the main results of this
section.
\begin{theorem}\label{Th:3d}
We consider the linearized system on the 3D torus $\mathbb{T}^3=[0,2\pi]^3$
with right-hand side $g_s$ and stationary solution $\vec u_0$ given in
\eqref{Kolmf} and \eqref{u0}, where
\begin{equation}\label{lambda3}
\lambda=\lambda_3(s)=\sqrt{2}\lambda_2(s,\gamma)=\sqrt{2}c_1\gamma\frac{(1+\alpha s^2)^2}s.
\end{equation}
Then for each triple of integers $a,b,r$ satisfying
\begin{equation}\label{abr}
\aligned
c_3s\le\widehat a=&\sqrt{a^2+b^2}\le c_4s, \quad |r|\le c_2s ,\\
&a\ge|b|,
\endaligned
\end{equation}
there exists an unstable solution of the linearized operator~\eqref{linear}.
\end{theorem}
\begin{proof}
We fix $a,b,r$ satisfying~\eqref{abr}.
Then in view of the first two inequalities in \eqref{abr}
the pair $(t',r)\in D\subset A(\delta)$, where $t'=\widehat a\cdot 1$
(so that we set $t=1$).
Applying Squire's transformation we obtain a 2D linearized problem on
the torus $\mathbb{T}^2_{\widehat a }$ of the form~\eqref{L2}
with $\widehat \gamma=\gamma\widehat a/a$. Then  in view of the third
inequality in  \eqref{abr} we have
$$
\lambda=
\sqrt{2}\lambda_2(s, \gamma)=
\lambda_2(s,\sqrt{2} \widehat \gamma\, a/\widehat a)\ge
\lambda_2(s,\widehat \gamma).
$$

We now see from  Proposition~\ref{Prop:eps} that
the 2D linearized problem~\eqref{L2} has an unstable
eigenvalue. Lemma~\ref{L:Squire} says, in turn, that so does
the linearized problem~\eqref{L3} on the standard torus $\mathbb{T}^3=[0,2\pi]^3$.
\end{proof}

The number of integers $(a,b,r)$ satisfying~\eqref{abr}
is of order $c_5s^3$, where
$$
c_5=\frac14\pi c_2(c_4^2-c_3^2).
$$

\begin{theorem}\label{Th:3d-lbd}
Let the right-hand side in~\eqref{DEalpha} be $g_s$ defined in~\eqref{Kolmf}
with $\lambda(s)$ defined in \eqref{lambda3}. Then the dimension of the
corresponding attractor $\mathscr{A}=\mathscr{A}_s$
satisfies for an  absolute constant $c_6$ the lower bound
\begin{equation}\label{3d-lbd}
\dim_F\mathscr{A}\ge c_6\frac{\| g_s\|_{L^2}^2}{\alpha^{5/2}\gamma^4}\,.
\end{equation}
\end{theorem}
\begin{proof}
We study our system in the limit  $\alpha\to 0$. Since $s$ is at our disposal we~set
$$
s=\frac1{\sqrt{\alpha}}\,.
$$
Then $\lambda$ in \eqref{lambda3} and  $\|g_s\|_{L^2}^2$ in~\eqref{Kolmf}
become
$$
\lambda=c_7\gamma{\sqrt{\alpha}},\qquad
\|g_s\|_{L^2}^2=c_8{\gamma^4}\alpha.
$$
We can finally write
$$
\dim_F\mathscr{A}\ge c_5s^3=c_5\frac1{\alpha^{3/2}}=
c_5\frac{\alpha \|g_s\|_{L^2}^2}{\alpha^{5/2}\|g_s\|_{L^2}^2}=
c_5\frac{\alpha \|g_s\|_{L^2}^2}{\alpha^{5/2}c_8\alpha\gamma^4},
$$
which gives~\eqref{3d-lbd} with $c_6=c_5/c_8$.
\end{proof}
\begin{remark}\label{Rem:lower2d}
{\rm
We recall  the sharp lower bound
in~\cite{IZLap70}  for the attractor dimension
for system~\eqref{DEalpha} on the 2D torus $\mathbb{T}^2$:
\begin{equation}\label{lower2d}
\dim_F\mathscr{A}\ge c_{\mathrm{abs}}
\frac{\|\operatorname{curl}g_s\|_{L^2}^2}{\alpha\gamma^4}\,.
\end{equation}
Since for $\lambda\sim\gamma{\sqrt{\alpha}}$ we have
$\|\operatorname{curl}g_s\|_{L^2}^2\sim\gamma^4$
and $\|g_s\|_{L^2}^2\sim{\gamma^4}\alpha$,
it follows that estimate~\eqref{lower2d} can equivalently be written
in terms of the  other dimensionless number in \eqref{0.est4} as follows
$$
\dim_F\mathscr{A}\ge c'_{\mathrm{abs}}
\frac{\|g_s\|_{L^2}^2}{\alpha^2\gamma^4}\,.
$$
}
\end{remark}

\setcounter{equation}{0}

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Pumpkin, spice and everything nice come together in this recipe for Pumpkin Chia Chai Overnight Oats.
Though it tastes like a decadent dessert, these overnight oats are a healthy way to start your day.
"Overnight oats are the perfect breakfast option for busy families on the go," says Rebecca Izawa, a registered dietitian at Kapiolani Medical Center for Women & Children.
When purchasing pre-canned pumpkin, make sure to pick up "pure pumpkin puree"; cans labeled "pumpkin pie filling" contain added sugar, which isn't so sweet.
Not these overnight oats! In fact, they're dairy-free, gluten-free and vegan, making them ideal for people with allergies or food sensitivities.
1. Place the oats, chia seeds, pumpkin puree, spices and maple syrup in a medium bowl and stir to combine. Add the almond milk and stir until incorporated. Cover with plastic wrap and chill overnight.
2. Divide mixture into four bowls. Top with chopped walnuts and pumpkin seeds and enjoy!


Brussels, 20 March - The European transport commissioner, Slovenia's Violeta Bulc, confirmed in Brussels on Wednesday that she would be a member of the campaign team of the Alliance of Liberals and Democrats for Europe (ALDE) for the European elections. She will not run herself in the May elections.
The news item consists of 2.943 characters (without spaces) or 592 words words.


ROAD safety, school run parking, health and sustainable travel were among the themes at an event for school pupils and parents.
Montrose Primary School in Aylestone hosted the Safe Happy Families event, at which families were able to learn more about local issues including inconsiderate parking, as well as finding out about local green transport initiatives, road safety, health and fitness.
The event involved representatives from Leicester City Council alongside cycling charity Sustrans, Living Streets and Leicestershire Police.
Visitors were able to learn more about the work of City Wardens and civil enforcement officers on tackling problems of school run parking, as well as discovering the Living Streets work to encourage more children to walk to school.
Health and fitness advice was also on offer, along with a chance to showcase the results of pupils' eco-themed activities, or explore the workings of a police camera enforcement vehicle.
More than 230 parents and guardians attended the event, which took place on Tuesday, September 25.
It also saw the launch of the Choose How You Move app schools competition, in which parents can earn points for logging sustainable travel, as well as earning reward points for the school by walking or cycling.
Aylestone ward councillor and deputy city mayor, Adam Clarke, was among those attending the event.
He said: "This event has given parents the chance to find out more about the work going on to help make their area safer and cleaner, as well as learning about alternative ways to travel to school.


text:
Calhoun debate shapes freshmen's first days at Yale
Tyler Foggatt & Emma Platoff 3:11 am, Sep 02, 2015
Staff Reporters
When Charles Kenney '19 walked into Woolsey Hall Saturday morning for the Freshman Assembly, he was expecting a standard college welcome address. Instead, he — and his 1,363 classmates — were issued a challenge.
In their speeches this weekend, University President Peter Salovey and Yale College Dean Jonathan Holloway called for the class of 2019's input on the controversial name of Calhoun College, which resurfaced as a topic of debate this summer following the removal of the Confederate flag from South Carolina's state capitol. Though most of the audience had been on campus for less than a full day, Salovey asked that the freshmen join the rest of campus in an open discussion regarding Calhoun's contentious name.
In the following days, University officials formally invited the entire Yale community to contribute to the debate, launching a website called "An Open Conversation" following the address. The website includes a submissions box for individuals to send in their opinions on the issue, as well as a list of upcoming public discussions.
"Members of the class of 2019, here is your first hard problem," Salovey said. "Welcome to Yale!"
During his speech, Salovey described the tragic massacre that took place in Charleston, South Carolina in June. He said the shootings ignited an impassioned national conversation about Confederate figures and symbols, a conversation that has also reached the University. Over the summer, members of the Yale community expressed dissatisfaction with the controversial namesake of Calhoun College: John C. Calhoun, a member of the class of 1804 who was a white supremacist and fierce supporter of slavery.
"Alumni and faculty have written to me and to Dean Holloway from varying perspectives, some at length and with considerable force," Salovey said. "And inevitably we found ourselves wondering, and not for the first time, how best to address the undeniable challengers associated with the fact that Calhoun's name graces a residential community in Yale College, an institution where, above all, we prize both the spirit and reality of full inclusion."
The call to action came as a surprise to many freshmen who were expecting, as Kenney put it, "basically just an inspirational speech about how smart Yale students are."
But most students interviewed said they appreciated being included in the important campus discussion so early in their Yale careers.
"Above all, I liked that they made it a discussion," Franklin Eccher '19 said.
And the discussion has already begun. According to Salovey, at least 500 responses have been submitted to the webpage form since it debuted on Saturday. Much of the input has been "incredibly thoughtful and deep," Salovey told the News, with messages coming from all parts of the Yale community and beyond.
He added that the responses have been "on all sides of the issue," and that it would be difficult to judge whether the majority of the responders are leaning one way or another.
"I have been impressed at the way the community has gotten engaged, and at people's willingness to be really thoughtful about what they have to say," Salovey said. "I was hoping for this and was confident that Yale is the kind of community where you can have a conversation like this."
When asked if a University consensus on the issue would lead to a change in the residential college's name, Salovey said that it is still hard to tell where the open discussion will lead, noting that the Yale Corporation has ultimate power over all campus buildings' names.
All nine freshmen interviewed said they were pleased that the issue had been raised in such a prominent forum.
Of those nine, five freshmen said they do not believe the name of the college should be changed, while two advocated for change and two said they are still struggling to decide. Most students against the change cited the robust tradition of the college's name and the disjunct between the man the college was named to honor and the experience of current Calhoun students.
"I agree with keeping the name … not because I agree with what he did, but because it's a part of history now," Katherine Melbourne '19 said. "We've obviously learned from our mistakes and are a very inclusive community now. The fact that it's named after him doesn't change who we are now."
These freshmen's positions clash with those of nearly 1,500 students and alumni who have signed a petition created by a group of Yale Law School students earlier this summer calling upon the University to change the name of Calhoun College.
With ongoing campus debate on the name of Calhoun College and a distinct but related conversation centering on the title of "master," this fall is an unusual time to adjust to life on campus. But most freshmen interviewed said that, far from being overwhelmed, they were actually glad to be coming to campus at such an important moment in the University's history.
"When we came onto campus, I had the idea that campus would kind of slow down and we'd be welcomed, then campus life would kind of start up again," Lily Mirfakhraie '19 said. "That hasn't been my experience at all. Campus and Yale are going in full swing, and we've been pushed into it … It's been a really good introduction."
TYLER FOGGATT
EMMA PLATOFF

redpajama_set_name: RedPajamaCommonCrawl


Holy crap, these things should have been around YEARS ago, and they certainly should be in stores now. Erasable highlighters! I got these as a gift from a friend of mine who shares my obsession with office supplies and they blew my freakin' mind.
Technologically speaking, they're nothing special — my sister and I had erasable markers as kids that worked exactly the same way — but in this GTD, DIY world of ours, oh, the uses! Let's say you're a student, for example. You want to highlight passages in a book to use in a term paper, rather than typing/writing them all out — but the book belongs to the library, or you want to sell it back at the end of the semester. Problem solved! Just erase all of your highlighting when your paper is finished. You could also highlight a to-do list chromatically according to priority and erase the highlighter when you cross off the things you've completed so they don't distract your eyes.
Of course, there are other, artsier things you could do with erasable highlighters, too, but from a business standpoint I think they're pretty darned useful. The only drawback I've found so far is that once you've erased a spot you can't re-highlight it, which means you can't change the color something is highlighted in. I have yet to see these in any office supply/stationery stores, though they supposedly carry them at Office Depot. You can find them here, at $2.99 for the whole set of four (plus shipping, natch).


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Two months back, rock icon David Bowie surprised fans by releasing his first single in a decade. The single came from his soon-to-be-released (March 12 if the buzz is true) new album The Next Day. iTunes is gracing us with some good old fashioned generosity and has made the full 53-minute album available for streaming. For the full listening experience, click here.
Bowie's 2013 release is an absolute treat for fans who had begun to assume, after his long silence, that Bowie was either in poor health or done with music. Since the release of his 2003 album Reality, Bowie has stayed almost entirely out of the public eye. On his 66th birthday on January 8, Bowie resurfaced with his new single "Where Are We Now?" and the promise of The Next Day.
Give The Next Day a listen and share your thoughts in the comments section. Is classic Bowie back?


First Lady Auxillia Mnangagwa ordered a Member of Parliament and a traditional chief removed from an event she had organised, ZimLive can reveal.
Mnangagwa recently spent two weeks in New York, the United States, where she travelled five days ahead of a United Nations summit on women, it is reported at the invitation of Women's Affairs Minister Sithembiso Nyoni.
She travelled to the province of her birth, Mashonaland Central, on Wednesday seeking a greater influence for herself in Mazowe Central constituency.
ZimLive understands enroute to the constituency where she wanted to identify projects she could support, Mnangagwa demanded to know who would be occupying the VIP tent with her.
On being advised that local MP Sydney Chidamba was present as well as Chief Lucious Chitsinde Negomo, she advised her security aides that she did not want to see the two.
"She gave instructions that the two must have nothing to do with her visit. The first lady did not want to see them. Chidamba and Chief Negomo were chucked out of the event, and they vowed never to attend her programmes," a security source briefed on the event told ZimLive.
Chidamba is the brother to former Cabinet minister Nicholas Goche's wife. The MP was previously married to former first lady Grace Mugabe's sister, Rose Chidhakwa.
"Goche and Grace Mugabe of course are seen as lynchpins of the so-called G40 faction in Zanu-PF. The first lady blames the chief and Chidamba of participating in previous attempts to block her from running for MP in that constituency," the source added.
President Emmerson Mnanagwa's wife is seen actively trying to sideline Chidamba, with speculation in the province that she wants to pave way for her brother Paradzai Kutyauripo – recently appointed director of state residencies – to capture the constituency in the next elections in 2023.
The first lady visited Negomo Irrigation Scheme for a general assessment. The scheme is in her ward of birth, and she told locals she was keen to see how she can assist farmers. She also toured Rosa Clinic, her birth place, and donated medical supplies and food items.
State media reported that Mnangagwa "had to cut short her attendance at the 63rd UN Commission on the Status of Women on Monday to assist citizens affected by Cyclone Idai that has so far claimed 139 lives in Zimbabwe."
Chidamba declined to comment for this story, while Chief Negomo could not be reached for comment.


text:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using AddressManager.Database;
using AddressManager.Models;
using AddressManager.Repositories;
using MongoDB.Driver;
using SchwabenCode.MongoDBRepository;

namespace AddressManager
{
    class Program
    {
        private const String MongoServer = "localhost";
        private const Int32 MongoPort = 27017;
        private const String MongoDbName = "AddressBook";
        private const String MongoUser = "UserName";
        private const String MongoPassword = "UserName";

        static void Main( String[ ] args )
        {
            // Default behavior, see http://docs.mongodb.org/ecosystem/tutorial/authenticate-with-csharp-driver/
            var credential = MongoCredential.CreateMongoCRCredential( MongoDbName, MongoUser, MongoPassword );

            var settings = new MongoClientSettings
            {
                //Credentials = new[ ] { credential },
                Server = new MongoServerAddress( MongoServer, MongoPort ),
                WriteConcern = WriteConcern.Acknowledged /* you can use fire and forget, too! */
            };

            var mongoClient = new MongoClient( settings );

            // Connect to Server
            var server = mongoClient.GetServer( );

            // Get our Database
            var database = server.GetDatabase( MongoDbName );

            // Create our IUoW Container
            IMongoUnitOfWork uowContainer = new DbContext( database );

            // ######### Examples - Addresses
            AddressRepository addressRepository = new AddressRepository( uowContainer );

            // ### Get All
            // Read all addresses with full entity details (uses default repository type)
            var addressesFull = addressRepository.GetAll( );
            // or with type
            var addressesFullByType = addressRepository.GetAll<Address>( );
            // or only specific fields
            var addessLabels = addressRepository.GetAll<AddressLabel>( );

            if ( addressesFull != null && addressesFullByType != null && addessLabels != null )
            {

            }

            // ######### Examples - Persons
            PersonRepository personRepository = new PersonRepository( uowContainer );

            // ### Write with validation
            var person = new Person { Name = "Test Person", EMail = "this would cause an expcetion" };
            WriteConcernResult addResult = null;
            try
            {
                addResult = personRepository.Add( person );
            }
            catch ( MongoInvalidEntityException e )
            {
                Console.WriteLine( "Failed to write." );
                foreach ( var error in e.Errors )
                {
                    Console.WriteLine( ">> " + error.ErrorMessage + " Affected fields: " + String.Join( ", ", error.MemberNames ) );
                }
            }

            // if MongoInvalidEntityException is catched, we have not result information and 'addResult' is null!
            if ( addResult == null )
            {
                Console.WriteLine( "No result information (invalid?? safemode off??)" );
            }
            else if ( addResult.Ok )
            {
                Console.WriteLine( "Successfully added person" );
            }
            else
            {
                Console.WriteLine( "Failed to write person: " + addResult.ErrorMessage );
            }

            Console.Read( );
        }
    }
}

redpajama_set_name: RedPajamaGithub