Équations de Navier-Stokes couplées à des équations de la chaleur: Résolution par une méthode de point fixe en dimension infinie. (Navier-Stokes equations connected with heat equations: Solution by a fixed point method in infinite dimension). (French) Zbl 0716.35064

The authors consider the following system of partial differential equations that results from combining the Navier-Stokes equations with heat equations: \[ (*)\quad div u=0,\quad -\mu \Delta u+(u\cdot \nabla)u+\nabla p=g(s)\text{ in } \Omega,\quad -\nu \Delta s+u\cdot \nabla s=0, \] together with Dirichlet conditions for u and mixed boundary conditions for s. Here \(\Omega \subset {\mathbb{R}}^ 2\) is a bounded open set, \(u=(u_ 1,u_ 2)\) and \(s=(s_ 1,...,s_ N)\) (N\(\geq 1).\)
For the subsequent reasoning it is assumed that \(\Omega\) is a two dimensional rectangle in which case precise estimates for the constants in Poincaré’s inequality and an interpolation inequality are given. This knowledge is used to give explicit conditions under which existence and uniqueness of a solution to (*) can be shown. The existence part is proved with the help of a fixed point theorem that is closely connected with the theory of monotone operators.
Reviewer: G.Dziuk


35Q30 Navier-Stokes equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
47H10 Fixed-point theorems
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