  This work studies the system of $3D$ stationary Navier-Stokes equations.
Several Liouville type theorems are established for solutions in mixed-norm
Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show
that, under some sufficient conditions in mixed-norm Lebesgue spaces, solutions
of the stationary Navier-Stokes equations are identically zero. This result
covers the important case that solutions may decay to zero with different rates
in different spatial directions, and some these rates could be significantly
slow. In the un-mixed norm case, the result recovers available results. With
some additional geometric assumptions on the supports of solutions, this work
also provides several other important Liouville type theorems for solutions in
weighted mixed-norm Lebesgue spaces. To prove the results, we establish some
new results on mixed-norm and weighted mixed-norm estimates for Navier-Stokes
equations. All of these results are new and could be useful in other studies.

Analysis of PDEs