Remarks on the strong solvability of the Navier-Stokes equations. (English) Zbl 1060.35102

Amann, H.

Funct. Differ. Equ. 8, No. 1-2, 3-9 (2001).

The author considers the Navier-Stokes equations \[ \begin{alignedat}{2} \nabla\cdot v= 0,\quad &\text{in }\Omega,\qquad &\partial_tv+ (v\cdot\nabla)v- \nu\Delta v= -\nabla p\quad &\text{in }\Omega,\\ v= 0\quad &\text{on }\partial\Omega,\qquad &v(\cdot,0)= V^0(x)\quad &\text{in }\Omega, \end{alignedat} \] where either \(\Omega= \mathbb{R}^m\), or \(\Omega\) is a half-space of \(\mathbb{R}^m\), or \(\Omega\) is a smooth domain in \(\mathbb{R}^m\), \(m\geq 3\). He establishes a relation between the maximal strong solution and Leray-Hopf weak solutions. He considers more general domains and non-vanishing exterior forces as well.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids

Keywords:

maximal strong solution; Leray-Hopf weak solutions; exterior forces