Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces
Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland · Calculus of Variations and Partial Differential Equations · 2022
Abstract This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an $$L^\infty $$ L ∞ -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in $${\mathbb {R}}^{n\times n}$$ R n × n with zero matrix traces. We analyze, in $$L^2$$ L 2 -based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in $${\mathbb {R}}^n$$ R n , $$n\ge 2$$ n ≥ 2 ( $$n=2,3$$ n = 2 , 3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.