Variational approach for the Stokes and Navier-Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds
Let \(M\) be an infinitely smooth, compact, boundaryless, oriented Riemannian manifold of dimension \(m\geq 2\). Let \(\Omega_+ \subset M\) be a domain with Lipschitz boundary such that \(\Omega_-: =M\setminus \overline \Omega_+\) is connected and non-empty. Let \(\mu \in L^\infty (M)\) be such that \(C_\mu^{-1}\leq \mu \leq C_\mu \) in \(M\) for some positive constant \(C_\mu \). Define \(L_\mu (u,\pi ):=\mathrm{Def}^* (2\mu \mathrm{Def}(u))+d\pi \). The paper is devoted to several problems corresponding to the system \(L_\mu (u,\pi )=f\), \(\delta u=0\). First the transmission problem in \(H^1(\Omega_\pm ,\Lambda^1TM) \times L^2(\Omega_\pm )\) is studied. Then the authors study the Dirichlet problem in \(H^1(\Omega_+ ,\Lambda^1TM)\times L^2(\Omega_+ )\). As a consequence, the Dirichlet problem is studied for the Navier-Stokes system \(L_\mu (u,\pi )+ \nabla_u u=f\), \(\delta u=0\), in \(H^1(\Omega_+ ,\Lambda^1TM)\times L^2(\Omega_+ )\).
scientific article; zbMATH DE number 6954519
scientific article