Well-Posedness of the Cauchy Problem for Systems

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 u _t + f(u) _x = 0, 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 |u( · , t) − v( · , t)|₁ ≤ C|u ₀ − v ₀|₁ for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, any weak solution coincides with the solution constructed by front tracking.

Ma per seguir virtute e conoscenza.¹ Dante Alighieri (1265–1321), La Divina Commedia