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About the concept of measure-valued solutions to distributed parameter systems. (English) Zbl 0828.35024

The aim of the paper is to investigate the concept of measure valued solutions in some partial differential equations. The usefulness of such solutions is related to the possibility to describe oscillations (typically in the gradient), but many definitions in the literature are not selective enough. In the paper some examples related to conservation laws, fluid dynamics, backward-forward heat problems are investigated. The nonlinear parabolic PDE \[ {\partial u \over \partial t} = \nabla \cdot \bigl( \varphi' (\nabla u) \bigr), \qquad \varphi : \mathbb{R}^n \to \mathbb{R}^n\tag \(*\) \] exhibits the behaviour of a forward heat equation in the regions \(\{x : \nabla u(x) \in \Omega^c\}\), \(\Omega^c\) being the convexity set of \(\varphi\), while it exhibits the behaviour of a backward heat equation in the regions \(\{x : \nabla u(x) \in \Omega_c\}\), \(\Omega_c\) being the concavity set of \(\varphi\). In the paper it is proposed a notion of measure valued solutions of the PDE \((*)\), and existence and uniqueness results are established.
Reviewer: L.Ambrosio (Pisa)

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35K05 Heat equation
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