Quantitative transfer of regularity of the incompressible Navier-Stokes equations from ${\mathbb{R}}^3$ to the case of a bounded domain

Abstract

Let $u_0\in C_0^5(B_{R_0})$ be divergence-free and suppose that $u$ is a strong solution of the three-dimensional incompressible Navier-Stokes equations on $[0,T]$ in the whole space ${\mathbb{R}}^3$ such that $\Vert u{\Vert }_{L^{\infty }((0,T);H^5({\mathbb{R}}^3))}+\Vert u{\Vert }_{L^{\infty }((0,T);W^{5,\infty }({\mathbb{R}}^3))}\le M<\infty$. We show that then there exists a unique strong solution $w$ to the problem posed on $B_R$ with the homogeneous Dirichlet boundary conditions, with the same initial data and on the same time interval for $R\ge \max (1+R_0,C(a)C(M{)}^{1/a}\exp (C{M}^{4}T/a)))$ for any $a\in [0,3/2)$, and we give quantitative estimates on $u-w$ and the corresponding pressure functions.

arXiv comment16 pages

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