Nonlinear semigroups and a hyperbolic conservation law

This paper is concerned with the hyperbolic conservation law M i\ Sxa . db(u) _ St + ^ x ^ ~ ° (1.2) u(x,O) = f (x) . By means of the Crandall-Liggett theory of nonlinear semigroups ([1]), it will be shown that the operator A : u H-> ——b(u) Q. X generates a semigroup S = {Sfc; t ;> 0} of contraction operators on L (IR). The function t h-» S.f may then be thought of as a generalized solution of the initial value problem (1.1-,2), and it will be seen that for f e L fl L 9 this function is a weak solution in the usual sense. Finally, semigroup methods will be employed to derive the "ordering principle" for the solutions of (1.1-.2), in a form due to Kruzkov [3]. 1 NONLINEAR SEMIGROUPS AND A HYPERBOLIC CONSERVATION LAW by H. Flaschka §1. Ihis paper is concerned with the hyperbolic conservation law (1.2) u(x,O) = f (x) . By means of the Crandall-Liggett theory of nonlinear semigroups ([1]), it will be shown that the operator A : u H-» -=—_:b(u) Q. X generates a semigroup S = {S ; t ̂ 0} of contraction operators on L (IR) . The function t i—» S.f may then be thought of as a generalized solution of the initial value problem (1.1-.2), and it will be seen that for f e L 0 L , this function is a weak solution in the usual sense. Finally, semigroup methods will be employed to derive the "ordering principle" for the solutions of (1.1-.2), in a form due to Kruzkov [3]. Ihe statement of the Crandall-Liggett theorem, and an overview of the technical aspects of the paper, will be given in §2. Ihe remainder of this introductory section is devoted to a general comparison of the present approach and the existing theory of equation (1.1). Ihe conservation law (1.1) has been studied quite thoroughly, both because of its mathematical interest, and because it This work was supported by NSF Grant GU-2056. is the prototype of some important hyperbolic systems describing physical phonomena. First of all, simple examples show that solutions of (1.1) can develop discontinuities ("shocks") even for smooth data (1.2); consequently, a solution can only be expected to satisfy the equation in a weak sense: (1...3) Definition. A bounded, measurable function u of (x,t) is said to be a weak solution of (1.1) , (1.2) , if i) u(t,x) —-> f(x) for a.e. x, as t —» 0; ii) fucp, + b(u)cp }dxdt = 0 for all twice cont>0 xeIR tinuously differentiable functions 0. The fundamental result in the theory, and the source of most of the complications, is this: 0 and all x,y e D(A) ,