The Formation of Shock Waves in Smooth Solutions

Abstract

From Corollary 2 of Theorem 2.2 in Chapter 2, we have learned that for smooth solutions u(x,t) of the system of conservation laws,

$$ \frac{{\partial u}}{{\partial t}} + \sum\limits_{j = 1}^N {\frac{\partial }{{\partial {x_j}}}{\mkern 1mu} {F_j}(u){\mkern 1mu} = {\mkern 1mu} 0} ,u(x,0) = {u_0}(x) $$

(3.1)

with initial data, u_o ∈ H^s(R^N), s > N/2 + 1 or more generally, u₀ ∈ H ^S_ul , two precise alternatives occur.