A short note on the Liouville problem for the steady-state Navier-Stokes equations

Abstract

Uniqueness of the trivial solution (the zero solution) for the steady-state Navier-Stokes equations is an interesting problem who has known several recent contributions. These results are also known as the Liouville type problem for the steady-state Navier-Stokes equations. In the setting of the $L^p-$ spaces, when $3\leq p \leq 9/2$ it is known that the trivial solution of these equations is the unique one. In this note, we extend this previous result to other values of the parameter $p$. More precisely, we prove that the velocity field must be zero provided that it belongs to the $L^p -$ space with $3/2<p<3$. Moreover, for the large interval of values $9/2<p<+\infty$, we also obtain a partial result on the vanishing of the velocity under an additional hypothesis in terms of the Sobolev space of negative order $\dot{H}^{-1}$. This last result has an interesting corollary when studying the Liouville problem in the natural energy space of these solutions $\dot{H}^{1}$.