TAILIEUCHUNG - Colombeau's theory and shock wave solutions for systems of PDEs

The modern notion of ecosystem services can be traced back to at least the early 1970s. Since the late 1990s, however, several well-known studies have codified ecosystem services into generally accepted lists or typologies (Daily 1997; DeGroot, Wilson et al. 2002) The Millennium Ecosystem Assessment (Millennium Ecosystem Assessment 2002; Mooney, Cropper et al. 2004; Pereira, Queiroz et al. 2005) classified ecosystem services into “supporting services,” the ecological processes and functions that generate other ecosystem services, “regulating services” that maintain global and local conditions at levels appropriate for human survival, “provisioning services” that offer physical resources directly contributing to human well-being,. | Electronic Journal of Differential Equations Vol. 2000 2000 No. 21 pp. 1-17. ISSN 1072-6691. URL http or http ftp ftp login ftp Colombeau s theory and shock wave solutions for systems of PDEs F. Villarreal Abstract In this article we study the existence of shock wave solutions for systems of partial differential equations of hydrodynamics with viscosity in one space dimension in the context of Colombeau s theory of generalized functions. This study uses the equality in the strict sense and the association of generalized functions that is the weak equality . The shock wave solutions are given in terms of generalized functions that have the classical Heaviside step function as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function that have to satisfy part of the equations in the strict sense and part of the equations in the sense of association. Introduction Let R R u x . Fix a p in R X R with a p. Let V be a function in C1 Rị 3 a p satisfying some conditions to be introduced in 5. We consider two associated systems of hydrodynamic equations with viscosity V in one space dimension. The system S consists of the equations Pt pu x 0 pu t p pu2 x v o p p e - -I et e p u x v o p p e - a uux x e Ap 2pu2 A 2 R and S consists of the two last equations and pt pu x 0 pu t p pu2 x v o p p e - a ux x where p is the density u the velocity p the pressure and e the total energy. The symbol denotes the association relation in Gs R2 R see 2 . The purpose of this paper is to study the existence of shock wave solutions see 5 for the systems S and S . More precisely solutions with two constant states separated Mathematics Subject Classifications 46F99 35G20. Key words and phrases Shock wave solution Generalized function Distribution. 2000 Southwest Texas State University and University of North Texas. Submitted January 13 2000. Published March

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