Meyers type estimates for the solution of a nonlinear Stokes system. (English. Russian original) Zbl 0911.35089

Zhikov, V. V.

Differ. Equations 33, No. 1, 108-115 (1997); translation from Differ. Uravn. 33, No. 1, 107-114 (1997).

The paper deals with the nonlinear Stokes system \[ \text{div}\bigl(a(x)| Du|^{p(x)-2}Du-z+\rho I\bigr)=0,\quad\text{div} u=0,\quad u|_{\partial\Omega}=0, \] in a bounded Lipschitz domain \(\Omega\subseteq \mathbb{R}^N\), where \(u=(u_1,\dots,u_N)\) is the velocity vector, \(Du=(1/2)\{\partial u_i/\partial x_j+\partial u_j/\partial x_i\}\) is the symmetric part of the Jacobi matrix \(\nabla u\), \(\rho(x)\) is the pressure, \(I\) is the identity matrix, the coefficients \(a(x)\) and \(p(x)\) are measurable, \(0<\nu\leq a(x)\leq\nu^{-1}\), \(1<\alpha\leq p(x)\leq\beta<\infty\), and \(z=\{z_{ij}(x)\}\) is a given tensor, \(z_{ij}=z_{ji}\in L^1(\Omega)\). The author finds sufficient conditions ensuring that solutions of the above problem satisfy the Meyers estimates. There are some applications of these estimates to boundary value problems in the hydrodynamics of quasi-Newtonian fluids.

Reviewer: V.Chernyatin (Szczecin)

MSC:

35Q30	Navier-Stokes equations
76D07	Stokes and related (Oseen, etc.) flows
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

nonlinear Stokes system; Lipschitz domain; Jacobi matrix; Meyers estimate; quasi-Newtonian fluid