PUBLICACIONES

Sociedad Colombiana de Matemáticas:Publicaciones

Revista Colombiana de Matemáticas

Volumen 41 [1] (2007)Páginas 81--90

Conservation laws I: viscosity solutions

Jin Yan,Zhixin Cheng,Ming Tao
University of Science and Technology of China, Hefei

Resumen.En este artículo usamos el teorema de punto fijo de Brouwer-Schauder para obtener la existencia de soluciones locales de viscosidad suave al problema de Cauchy para el sistema parabólico \[ \left\{ \begin{aligned} &u^1_t+f_1\left(u^1,u^2,\cdots,u^n\right)_x+g_1\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^1_{xx} & \qquad \qquad \qquad \qquad \qquad \quad \vdots &u^n_t+f_n\left(u^1,u^2,\cdots,u^n\right)_x+g_n\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^n_{xx}, \end{aligned} \right. \] con data inicial medible acotada $$u^1(x,0)=u^1_0(x),\quad u^2(x,0)=u^2_0(x), \cdots, \quad u^n(x,0)=u^n_0(x).$$ Luego, basados en la existencia local y el principio del máximo, obtenemos la existencia de soluciones globales suaves para dos sistemas especiales , uno relacionado con el sistema parabólico de flujo cuadrático y el otro relacionado con el sistema LeRoux.

Abstract. In this paper we use the Brouwer-Schauder's fixed point theorem to obtain the existence of local smooth viscosity solutions of the Cauchy problem for the parabolic system \[ \left\{ \begin{aligned} &u^1_t+f_1\left(u^1,u^2,\cdots,u^n\right)_x+g_1\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^1_{xx} & \qquad \qquad \qquad \qquad \qquad \quad \vdots &u^n_t+f_n\left(u^1,u^2,\cdots,u^n\right)_x+g_n\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^n_{xx}, \end{aligned} \right. \] with the bounded measurable initial data $$u^1(x,0)=u^1_0(x),\quad u^2(x,0)=u^2_0(x), \cdots, \quad u^n(x,0)=u^n_0(x).$$ Then based on the local existence and the maximum principle, we get the existence of global smooth solutions for two special systems, one related to the hyperbolic system of quadratic flux and the other related to the LeRoux system.

Palabras claves. Hyperbolic conservation laws, viscosity solution, Cauchy problem, a priori estimate.

Codigo AMS. Primary: 35B40. Secondary: 35l65.

Archivo completo : Formato [PDF] (770 K).