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
Full Text:

References:

 [1] and , ’Proposed experimental tests of a theory of fine microstructure and the two-well problem’, a preprint. [2] Chipot, Arch. Rational Mech. Anal. 103 pp 237– (1988) [3] DiPerna, Arch. Rational Mech. Anal. 88 pp 223– (1985) [4] DiPerna, Comm. Math. Phys. 108 pp 667– (1987) [5] and , ’Thermomechanical evolution of a microstructure’, submitted. · Zbl 0815.49029 [6] and , ’Optimal control of a finite structure’, submitted. [7] Hollig, Trans. AMS 278 pp 299– (1983) [8] Illner, J. Math. Anal. Appl. 157 pp 351– (1991) [9] Kinderlehrer, SIAM J. Math. Anal. 23 pp 1– (1992) [10] Neustupa, Math. Methods in Appl. Sci. 14 pp 93– (1991) [11] ’A note about optimality conditions for variational problems with rapidly oscillating solutions’, in: Progress in Partial Differential Equations: Calculus of Variations, Applications, Eds. et al., Longmann, in print. [12] ’Evolution of a microstructure: a convexified model’, Math. Methods in Appl. Sci., in print. · Zbl 0804.49036 [13] ’Optimality conditions for nonconvex variational problems relaxed in terms of Young measures’, submitted. · Zbl 1274.49040 [14] and , ’Measure-valued solutions to a problem in dynamic phase transitions’, in Nonstrictly Hyperbolic Conservation Laws, Eds. and , Contemporary Math., 60, 115-124, AMS, Providence, 1987. [15] ’Measure-valued solutions to a backward-forward heat equation: a conference report’, in Nonlinear Evolution Equations that Change Type, Eds. and , IMA Vol.’ in Math. and Its Appl. 27, 232-242, Springer, New York, 1990. [16] Slemrod, J. Dyn Diff. Equations 3 pp 1– (1991) [17] ’Discontinuities and oscillations’, in Directions in Partial Differential Equations, Eds. and , pp. 211-233, Academic Press, New York, 1987. [18] Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Philadelphia, PA, 1969. · Zbl 0177.37801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.