Liouville type theorems for 3D stationary Navier-Stokes equations in\n weighted mixed-norm Lebesgue spaces

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