On a nonlinear parabolic problem arising in some models related to turbulent flows. (English) Zbl 0808.35066

Summary: This paper studies the Cauchy-Dirichlet problem associated with the equation \[ b(u)_ t- \text{div} (|\nabla u-K(b(u)) e|^{p-2} (\nabla u- K(b(u)) e))+ g(x,u)= f(t,x). \] This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media and gases flowing in pipes of uniform cross sectional areas. The paper focuses on the class of bounded weak solutions, and shows (under suitable assumptions) their stabilization, as \(t\to\infty\), to the set of bounded weak solutions of the associated stationary problem. The existence and comparison properties (implying uniqueness) of such solutions are also investigated.


35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
76S05 Flows in porous media; filtration; seepage
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