Chapter 1 contains what was originally published as [BLe]. In the framework of the first point above, we prove that every Lax admissible weak solution of (1) coincides with the corresponding SRS trajectory if and only if it has locally bounded variation along a suitable family of space-like curves. By the uniqueness of SRS, there follows a uniqueness result for (1) (2), within the class of solutions having the mentioned property. Chapters 2 and 3 are concerned with the L1 stability of wave patterns containing some non-interacting large shock waves. We study the problem (1) with u in (2) being a small L^ 1 ∩ BV perturbation of fixed Riemann data. We a priori assume that the solution of the latter problem is given by a number of (arbitrarily large) Lax compressive and Majda stable shocks of different characteristic families. We formulate the BV and 11 Stability Conditions that express the expected mutual influence of the large waves. By constructing suitable Glimm and Lyapunov functionals applicable to our setting, we show that the former condition guarantees the existence of a unique, global in time and space, 'admissible' solution to (1) (2); while the latter condition is essential for the stablity of this same solution under (a class of) perturbations of its initial data. This is carried out in Chapter 2, containing the results of (Lel]. In Chapter 3 we present a revised version of the article (Le2], with some new additions. Several authors had investigated the issue of wellposedness of (1) (2) in various contexts, introducing different stability conditions. Some of them require that the eigenvalues of suitable matrices related to wave transmissions - reflections are smaller than 1 in absolute value, other refer to different algebraic properties of the linearised system, such as for example existence of weights with whom the flow of the system becomes a contraction. We explain and compare these conditions, showing that the conditions of Chapter 2 generalize or unify them in appropriate ways.

Topics in the Stability of Systems of Conservation Laws(2000 Oct 20).

Topics in the Stability of Systems of Conservation Laws

-
2000-10-20

Abstract

Chapter 1 contains what was originally published as [BLe]. In the framework of the first point above, we prove that every Lax admissible weak solution of (1) coincides with the corresponding SRS trajectory if and only if it has locally bounded variation along a suitable family of space-like curves. By the uniqueness of SRS, there follows a uniqueness result for (1) (2), within the class of solutions having the mentioned property. Chapters 2 and 3 are concerned with the L1 stability of wave patterns containing some non-interacting large shock waves. We study the problem (1) with u in (2) being a small L^ 1 ∩ BV perturbation of fixed Riemann data. We a priori assume that the solution of the latter problem is given by a number of (arbitrarily large) Lax compressive and Majda stable shocks of different characteristic families. We formulate the BV and 11 Stability Conditions that express the expected mutual influence of the large waves. By constructing suitable Glimm and Lyapunov functionals applicable to our setting, we show that the former condition guarantees the existence of a unique, global in time and space, 'admissible' solution to (1) (2); while the latter condition is essential for the stablity of this same solution under (a class of) perturbations of its initial data. This is carried out in Chapter 2, containing the results of (Lel]. In Chapter 3 we present a revised version of the article (Le2], with some new additions. Several authors had investigated the issue of wellposedness of (1) (2) in various contexts, introducing different stability conditions. Some of them require that the eigenvalues of suitable matrices related to wave transmissions - reflections are smaller than 1 in absolute value, other refer to different algebraic properties of the linearised system, such as for example existence of weights with whom the flow of the system becomes a contraction. We explain and compare these conditions, showing that the conditions of Chapter 2 generalize or unify them in appropriate ways.
Lewicka, Marta
Bressan, Alberto
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/4268
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