DOI: http://dx.doi.org/10.18273/revint.v35n1-2017002

Articulo Original

Hyperbolicity and genuine nonlinearity conditions for certain p-systems of conservation laws, week solutions and the entropy condition

Condiciones de hiperbolicidad y no linealidad genuina para ciertos p-sistemas de leyes de conservación, soluciones débiles y condición de entropía

 

Edgardo Pérez1∗

Krzysztof Rózga2

 

1Université de Brest, Département de Mathématiques, Brest, France.

2University of Puerto Rico at Mayaguez, Department of Mathematical Sciences, Mayaguez, Puerto Rico.

 

E-mail: edgardomath@hotmail.com

 

Abstract.

We consider a p-system of conservation laws that emerges in one dimensional elasticity theory. Such system is determined by a function W. We consider four forms of W. These are St.Venant-Kirchhoff, Ogden, Kirchhoff modified, Blatz-Ko-Ogden forms. In each of those cases we determine the conditions for the parameters µ, λ and f, under which the corresponding system is hyperbolic and genuinely nonlinear. We also establish what it means a weak solution of an initial and boundary value problem. Finally we ask if such solutions satisfy the entropy condition. For a standard entropy function we provide a complete answer, except of the Blatz-Ko-Ogden case. For a general strictly convex entropy function the result is that for the initial value of velocity fun

Keywords: Weak solution, entropy condition, conservation laws, genuinely nonlinear, p-system.

 

Resumen.

Consideramos un p-sistema de leyes de conservación que surge de la teoría de elasticidad unidimensional. Tal sistema es determinado por una función W. Consideramos cuatro formas de W. Estas son los modelos de St.

Venant-Kirchhoff, Ogden, Kirchhoff modificado y Blatz-Ko-Ogden. En cada uno de estos casos determinamos las condiciones de los parámetros µ, λ y f bajo las cuales el correspondiente sistema es hiperbólico y genuinamente no lineal. Establecemos qué significa una solución débil de un problema de valor inicial y de frontera. Finalmente nos preguntamos si tales soluciones satisfacen la condición de entropía. Para una función de entropía estándar damos una respuesta completa, excepto del caso de Blatz-Ko-Ogden. Para una función de entropía general estrictamente convexa, el resultado es que para el valor inicial de la función velocidad cerca de cero estas soluciones satisfacen la condición de entropía, bajo la restricción de hiperbolicidad y no linealidad genuina.

Palabras clave: Solución débil, condición de entropía, leyes de conservación, genuinamente no lineal, p-sistema.

 

Received: 20 January 2017,

Accepted: 25 April 2017

 

Texto Completo disponible en PDF

 

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To cite this article: E. Pérez, K. Rózga, Hyperbolicity and genuine nonlinearity conditions for certain p-systems of conservation laws, week solutions and the entropy condition, Rev. Integr. Temas Mat. 35 (2017), No. 1, 11–36.