Abstract
In this paper, we study the convergence rates of solutions in homogenization of nonlinear Stokes Dirichlet problems. The main difficulty of this work is twofold. On the one hand, the nonlinear Stokes problems do not fit the standard framework of second-order elliptic equations in divergence form. On the other hand, nonlinear problems may cause new difficulties in the estimation of the quantity as well as first-order approximate term. As a consequence, we establish the sharp rates of convergence in H^{1} and L^{2}. This work may be regarded as an extension of the approach for the linear Stokes problems to the nonlinear case.
Highlights
1 Introduction The main purpose of this paper is to establish the sharp rates of convergence in H1 and L2 for nonlinear Stokes problems with the Dirichlet boundary condition
For the case of Stokes problems, some outstanding results about regularity and convergence of solutions in homogenization were established by Gu and Shen in a series of papers
The nonlinear Stokes problems do not fit the standard framework of second-order elliptic equations in divergence form, which is caused by the pressure term
Summary
1 Introduction The main purpose of this paper is to establish the sharp rates of convergence in H1 and L2 for nonlinear Stokes problems with the Dirichlet boundary condition. The periodic functions (N, χ ) ∈ H1(Rn) × L2(Rn) are the so-called correctors, satisfying the following cell problem: The existence and convergence results of the weak solution to problem (1.1) may be found in [4, 12]. There are many such classic works about convergence results of solutions in homogenization of second-order elliptic equations with the various settings.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
