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 which has known several recent contributions. The problem is also known as the Liouville problem for the steady-state Navier–Stokes equations. In the setting of the $L^p$-spaces, when $3\le p \le 9/2$, 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 $\dot{H}^{1}$.