An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions.
The classical Navier-Stokes system for incompressible fluid in a bounded domain \(\Omega \subset \mathbb R^ 3\) is coupled with heat equation to render thermal effects (Boussinesq approximation). The boundary \(\partial\Omega\) is split into 5 subsets \(\Gamma_j,\) where the Dirichlet boundary condition for velocity is given on \(\Gamma_1,\) normal component of stress tensor on \(\Gamma_2\) and Dirichlet, Neumann and Newton boundary condition on \(\Gamma_3, \Gamma_4\) and \(\Gamma_5,\) respectively. Initial data for velocity and temperature are prescribed. For given problem the existence and uniqueness of local (maximal) weak solution is proved by an application Schauder fixed point principle.
Sobolev space with non-integer order derivatives
scientific article; zbMATH DE number 1508451