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Liouville type theorems on the steady Navier-Stokes equations in R3 (1710.06569v2)
Published 18 Oct 2017 in math.AP, math-ph, and math.MP
Abstract: In this paper we study the Liouville type properties for solutions to the steady incompressible Navier-Stoks equations in $\mathbf{R}{3}$. It is shown that any solution to the steady Navier-Stokes equations in $\mathbf{R}{3}$ 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.