Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor

Mirela Kohr,Sergey E. Mikhailov,Wolfgang L. Wendland

doi:10.1080/17476933.2019.1631293

Abstract

ABSTRACTWe obtain well-posedness results in -based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with strongly elliptic coefficient tensor, in complementary Lipschitz domains of , . The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole to an equivalent mixed variational formulation with data in -based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space , , for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in -based weighted Sobolev spaces, whenever the given data are small enough.

Highlights

A powerful tool in the analysis of boundary value problems for partial differential equations is played by the layer potential methods
Choi and Lee [10] proved the well-posedness in Sobolev spaces for the Dirichlet problem for the Stokes system with nonsmooth coefficients in a Lipschitz domain ⊂ Rn (n ≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations (VMO) with respect to all variables
Choi and Yang [11] established existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in the space Rd, d ≥ 3, when the weak solutions of the system are locally Hölder continuous

Summary

Introduction

A powerful tool in the analysis of boundary value problems for partial differential equations is played by the layer potential methods. Dindos and Mitrea [7, Theorems 5.1, 5.6, 7.1, 7.3] used a boundary integral approach to show well-posedness results in Sobolev and Besov spaces for Poisson problems of Dirichlet type for the Stokes and Navier–Stokes systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. The authors in [9] used a layer potential approach and a fixed point theorem to show well-posedness of transmission problems for the Navier–Stokes and Darcy-Forchheimer-Brinkman systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. Brewster et al in [22] used a variational approach to show well-posedness results for Dirichlet, Neumann and mixed problems for higher order divergence-form elliptic equations with L∞ coefficients in locally ( , δ)-domains in Besov and Bessel potential spaces. The approaches based on the pseudostress formulation have been intensively used in the study of viscous incompressible fluid flows due to their ability to avoid the symmetry condition that appears in the approaches based on the standard stress formulation (see, e.g. [29, 30])

Standard Lp-based Sobolev spaces and related results

Abstract mixed variational formulations and well-posedness results

Poisson problem of transmission type for the anisotropic Stokes system

Auxiliary results: equivalent norms in Banach spaces