DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.23.185 DYNAMICAL SYSTEMS Volume 23, Number 1&2, January & February 2009 pp. 185–195 A VARIATIONAL APPROACH TO THE RIEMANN PROBLEM FOR HYPERBOLIC CONSERVATION LAWS Constantine M. Dafermos Division of Applied Mathematics Brown University, Providence, RI 02912, USA Abstract. Within the framework of strictly hyperbolic systems of conserva- tion laws endowed with a convex entropy, it is shown that the admissible solu- tion to the Riemann problem is obtained by minimizing the entropy production over all wave fans with fixed end-states. 1. Introduction. We consider strictly hyperbolic systems of n conservation laws in one spatial dimension: ∂ t U (x, t)+ ∂ x F (U (x, t)) = 0. (1.1) The state vector U takes values in R n . The flux F is a given smooth map from R n to R n , and for any U ∈ R n the Jacobian matrix DF (U ) has real distinct eigenvalues λ 1 (U ) < ··· <λ n (U ), (1.2) which are the characteristic speeds. The books [4][8][10][15] provide detailed expositions of the current state of the theory of such systems. When the flux is nonlinear, solutions starting out from even smooth initial values eventually develop jump discontinuities that propagate on as shock waves. Thus, the Cauchy problem in the large must be set in the realm of weak solutions. Furthermore, in order to secure well-posedness, one has to impose admissibility conditions, which are usually motivated by physics. The subject of this paper is the Riemann Problem, namely the Cauchy problem for (1.1) with initial data of the form U (x, 0) =    U L , x< 0 U R , x> 0, (1.3) where U L and U R are constant states. Since both (1.1) and (1.3) are invariant under uniform stretching of the spatial and temporal coordinates, this problem possesses self-similar solutions U (x, t)= V (x/t). Self-similar solutions play a central role, as they govern both the local structure and the large time behavior of general weak solutions, that is, figuratively speaking, they depict how solutions look under the microscope or through a telescope. More- over, solutions to Riemann problems serve as the building blocks for constructing BV solutions to the Cauchy problem by the random choice method [9] or by the front tracking algorithm [4]. 2000 Mathematics Subject Classification. Primary: 35L65. Key words and phrases. Hyperbolic conservation laws, Riemann Problem, Entropy. Supported by the NSF under grants DMS-0202888 and DMS-0244295. 185