Liouville type theorems on the steady Navier-Stokes equations in R3

2017/10/18 by Xin, Zhouping, Xu, Deliang
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1710.06569

Abstract

In this paper we study the Liouville type properties for solutions to the steady incompressible Navier-Stoks equations in R³. It is shown that any solution to the steady Navier-Stokes equations in R³ with finite Dirichlet integral and vanishing velocity field at far fields must be trivial. This solves an open problem. The key ingredients of the proof include a Hodge decomposition of the energy-flux and the observation that the square of the deformation matrix lies in the local Hardy space. As a by-product, we also obtain a Liouville type theorem for the steady density-dependent Navier-Stokes equations.

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