SOME EXISTENCE AND UNIQUENESS RESULTS FOR A TIME-DEPENDENT COUPLED PROBLEM OF THE NAVIER–STOKES KIND

Abstract

In this paper, we consider some systems which are close to the instationary Navier–Stokes equations. The structure of these systems is the following: An (N + 1)-dimensional equation for motion (including the incompressibility condition) and a scalar equation involving an additional unknown, k = k(x,t). Among other things, they serve to model the behavior of certain turbulent flows. We are mainly concerned with existence and uniqueness results. The main difficulties are due to the scalar equation. In particular, the right-hand side is typically in L1; furthermore, there are nonlinear terms of the kind ∇·(μ(k)∇ k) and ∇·(B(k)), where μ and B are general continuous functions (no growth condition at infinity is imposed). Following the previous work of other authors, it is crucial to introduce the notion of weak-renormalized solution. Our results give the existence in the two-dimensional case, as well as the uniqueness of regular solution in both the two- and three-dimensional cases.