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

Zhouping Xin; Deliang Xu

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arXiv · 1710.06569

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

Zhouping Xin·Deliang Xu

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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.

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Zhouping Xin, Deliang Xu. 2017-10-18. Liouville type theorems on the steady Navier-Stokes equations in R3. https://arxiv.org/abs/1710.06569

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