Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 2

On the higher integrability of weak solutions to the generalized Stokes system with bounded measurable coefficients

Pages: 127 – 146

DOI: http://dx.doi.org/10.4310/DPDE.2018.v15.n2.a3

Authors

P. Kaplický (Department of Mathematical Analysis, Charles University in Prague, Czech Republic)

J. Wolf (Department of Mathematics, Chung-Ang University, Seoul, South Korea)

Abstract

In this paper, we deal with the generalized Stokes and Navier–Stokes problem. The elliptic term in the equation is assumed to have form $- \mathrm{div} (AD (u))$, where the matrix function $A$ is uniformly positive definite, but only $L^{\infty}$. Using a Meyers’ type estimate we improve the integrability of gradients of local weak solutions to a generalized Stokes problem. We also show that in the case of planar motion the integrability of local weak solution to generalized Navier–Stokes system can be improved. This in combination with previous result gives better properties of gradient of solutions.

Keywords

Navier–Stokes equations, Generalized Stokes system, higher integrability, local regularity

2010 Mathematics Subject Classification

Primary 34M40, 35Q30. Secondary 35B65, 76B03.

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Received 4 November 2016