Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations

Abstract

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in ${\mathbb{R}}^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1({\mathbb{R}}^N)$ solution $(v,p)$ with $|v{|}^2+|p|\in L^{\frac{q}{2}}({\mathbb{R}}^N)$, $q\in (\frac{3N}{N-1},\infty )$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition ${\int }_{{\mathbb{R}}^3}|\Delta v{|}^{\frac{6}{5}}dx<\infty$, which is stronger than the important D-condition, ${\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx<\infty$, but both having the same scaling property, implies that $v=0$. In the appendix we reprove the Theorem 1.1(\cite{cha0}), using the self-similar Euler equations directly.

15 pages(to appear in Comm. Math. Phys.). arXiv admin note: substantial text overlap with arXiv:1105.3639

References in corpus (1)

• Nonexistence of self-similar singularities for the 3D incompressible Euler equations

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