Entropy solutions for linearly degenerate hyperbolic systems of rich type |
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Authors: | Ta-Tsien Li,Yue-Jun Peng,Jé ré my Ruiz |
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Affiliation: | aSchool of Mathematical Sciences, Fudan University, Shanghai 200433, PR China;bLaboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex, France |
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Abstract: | Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems. |
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Keywords: | Explicit solution Rich system Linearly degenerate characteristic Non-strict hyperbolicity Existence of entropy solution |
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