Dynamics of Partial Differential Equations

Volume 17 (2020)

Number 3

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

Pages: 229 – 243

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n3.a2


Tuoc Phan (Department of Mathematics, University of Tennessee, Knoxville, Tn., U.S.A.)


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 of 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.


Liouville type theorem, Navier–Stokes equations, mixed-norm Lebesgue spaces, weighted mixed-norm Lebesgue spaces, Muckenhoupt weights, extrapolation theory

T. Phan’s research is partially supported by the Simons Foundation, grant # 354889.

Received 17 May 2019

Published 7 July 2020