Existence and uniqueness results for a coupled problem related to the stationary Navier-Stokes system
The authors consider the stationary boundary value problem in a bounded domain \(\Omega\subset \mathbb{R}^n\) \[ -\nu\Delta u-\nabla\cdot(k\Phi (\nabla u))+(u\cdot\nabla )u +\nabla p=f,\quad\nabla\cdot u=0\tag{1} \] \[ -\nabla\cdot (\mu (k)\nabla k+B(k))+u\cdot\nabla k=\nu|\nabla u|^2 +k\Phi (\nabla u):\nabla u,\tag{2} \] \(u=0\), \(k=0\) on \(\partial\Omega\). Here the vector \(u\) and scalars \(p\) and \(k\) are unknown, \(\Phi, f,\mu,B\) are given functions \(\nu=\text{const}>0\), \(\mu\) and \(B\) are continuous functions. The system (1), (2) can be used to describe the behavior of certain turbulent flows. The system may be considered as a model of non-Newtonian mechanics. Namely, \(u\) is the velocity field, \(p\) is the pressure, \(k\) is the temperature, and the stress tensor \(T\) has a form \[ T=\nu S(u)+k\Phi(S(u)),\qquad S(u)=\nabla u+(\nabla u)^T \] The existence of a weak solution to the problem and under some additional conditions the uniqueness are proved.
scientific article; zbMATH DE number 1022297