JOURNAL OF DIFFERENTIAL EQUATIONS 80, 343-363 (1989) The Riemann Problem for 2 x 2 Systems of Hyperbolic Conservation Laws with Case I Quadratic Nonlinearities MICHAEL SHEARER* Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205 Received September 20, 1988 1. INTRODUCTION A 2 x 2 system of conservation laws U,+F(U),=O, (1.1) where F: IF!’+ lR2is a smooth function, has an umbilic point at U = U0 if dF( U,) is a multiple of the identity matrix. Suppose system (1.1) is strictly hyperbolic for nearby U # U,, (i.e., the eigenvalues of dF( U) are real and distinct). Then Eq. (1.1) is related to the normal form where U, + dC( U), = 0, (1.2) C(u, v) = au3/3 + bu2v + uv*, a # 1 + b*. (1.3) In [9], this connection between nonstrictly hyperbolic systems (1.1) with an isolated umbilic point and the normal form (1.2), (1.3) is established in detail, and a classification of the equations is given in terms of a and b. This paper completes the study of Riemann problems for 2 x 2 systems of nonstrictly hyperbolic conservation laws with quadratic nonlinearities that are nondegenerate in the senseof [9]. In [lo], the Riemann problem was solved for Eq. (1.2) with a > 36*/4. In this paper, the Riemann problem is solved for Eq. (1.2) with a < 3b2/4 (Case I of [9]) and arbitrary initial data w&O)= u { UL if x<O R if x > 0. (1.4) * Research supported by National Science Foundation Grant DMS-8701348 and Army Research Office Grant DAAL03-88-K-0080. 343 0022-0396189 $3.00 Copyright 0 1989 by Academic Press. Inc. All rights of reproduction in any form reserved.