Well-Posedness of the Cauchy Problem

Abstract

The goal of this chapter is to show that the limit found by front tracking, that is, the weak solution of the initial value problem

$$\displaystyle u_{t}+f(u)_{x}=0,\quad u(x,0)=u_{0}(x),$$

(7.1)

is stable in L ¹ with respect to perturbations in the initial data. In other words, if $v=v(x,t)$ is another solution found by front tracking, then

$$\displaystyle{\left\|u(\,\cdot\,,t)-v(\,\cdot\,,t)\right\|}_{1}\leq C{\left\|u_{0}-v_{0}\right\|}_{1}$$

for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, every weak solution coincides with the solution constructed by front tracking.

Ma per seguir virtute e conoscenza. — Dante Alighieri (1265–1321), La Divina Commedia Hard to comprehend? It means ‘‘[but to] pursue virtue and knowledge.’’

Ma per seguir virtute e conoscenza.

— Dante Alighieri (1265–1321), La Divina Commedia

Hard to comprehend? It means ‘‘[but to] pursue virtue and knowledge.’’