The equation
is the simplest model of nonlinear wave motion; here x and t are scalar quantities, f a given smooth map from the real line into itself and u(x, t ) a realvalued function. We shall study the initial value problem, that is, the relation of solutions u at time t > 0 to their initial value
This paper shows the equivalence of the so-called entropy con,&tion for Equation ( 1) is hyperbolic; carrying out the differentiation we get generalized solutions and the L,-contractiveness of the solution operator.
u, + f ' ( u ) u , = 0.
Clearly u(x, t ) is equal to uo(x) on the line x ( t ) satisfying dx/dt = f ' ( u ( x , t)) ; such lines are called characteristics. Since the slope of the characteristic depends upon the solution (unlike the characteristics of the linear wave. equation which have constant slope), characteristics may intersect, and where they do, no smooth solution will exist.
This phenomenon of nonexistence of smooth solutions corresponds to the development of shock waves in the physical problem of compressible fluid flow. Mathematically these can be described as curves x ( t ) across which u(x, t ) fails to be continuous, but such that u has a limit as points ( x , t ) approach the curve from either side. The shock curves must also respect the fact that equation * This research was carried out under AEC Contract number AT(30-1)-1480-V at the Courant Institute of Mathematical Sciences, New York University. Reproduction in whole or in part is permitted for any purpose of the United States Government.