Abstract
This paper is concerned with the proof of existence and numerical approximation of large-data global-in-time Young measure solutions to initial-boundaryvalue problems for multidimensional nonlinear parabolic systems of forward-backward type of the form ?tu - div(a(Du))+ Bu = F, where B ? Rmxm, Bv?v ? 0 for all v ? Rm, F is an m-component vector-function defined on a bounded open Lipschitz domain ? ? Rn, and a is a locally Lipschitz mapping of the form a(A)= K(A)A, where K: Rmxn ? R. The function a may have unequal lower and upper growth rates; it is not assumed to be monotone, nor is it assumed to be the gradient of a potential. We construct a numerical method for the approximate solution of problems in this class, and we prove its convergence to a Young measure solution of the system.
Highlights
The paper is concerned with the numerical approximation of Young measure solutions to initial-boundary-value problems for nonlinear multidimensional parabolic systems of forward-backward type, where the existence of a weak solution cannot in general be guaranteed, because the nonlinearity in the equation is Corresponding author
Nonlinear parabolic systems of this type arise in certain mathematical models of the atmospheric boundary layer and, to date, there have been no attempts at the rigorous mathematical analysis of numerical methods for their approximate solution
Our terminology Young measure solution follows that of Frehse & Specovius-Neugebauer in [12]; in particular, we have consciously avoided referring to the solutions considered here as measure-valued solutions, as the function u, whose existence we prove, is still a real-valued function with spatial Sobolev regularity; instead of being a standard weak solution, the function u satisfies the PDE in the sense of gradient Young-measures
Summary
The paper is concerned with the numerical approximation of Young measure solutions to initial-boundary-value problems for nonlinear multidimensional parabolic systems of forward-backward type, where the existence of a weak solution cannot in general be guaranteed, because the nonlinearity in the equation is Corresponding author. Young measure solutions to forward-backward parabolic problems were studied by Demoulini in [7]: under the assumptions that m “ 1, p “ q “ 2 and that (11) is satisfied, a sequence of approximating solutions was constructed by means of minimizing an integral, which involves the convexification of Φ Using this construction, a family of Young measures was generated displaying an independence property, from which it was possible to deduce a uniqueness result regarding the function u (not regarding the family of Young measures).
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