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This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data. The new results obtained here are sharp in low dimension. 1. Introduction 1.1. The results. We consider in this paper second order, nonlinear hyperbolic equations of the form (1.1) gij (u) ∂i ∂j u = q ij (u) ∂i u ∂j u on R × Rn , with Cauchy data prescribed at time 0, (1.2) u(0, x) = u0 (x) , ∂0 u(0, x) = u1 (x) . | Annals of Mathematics Sharp local well-posedness results for the nonlinear wave equation By Hart F. Smith and Daniel Tataru Annals of Mathematics 162 2005 291 366 Sharp local well-posedness results for the nonlinear wave equation By Hart F. Smith and Daniel Tataru Abstract This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data. The new results obtained here are sharp in low dimension. 1. Introduction 1.1. The results. We consider in this paper second order nonlinear hyperbolic equations of the form 1.1 gij u didju qij u diudju on R X Rra with Cauchy data prescribed at time 0 1.2 u 0 x u0 x d0u 0 x u1 x . The indices i and j run from 0 to n with the index 0 corresponding to the time variable. The symmetric matrix gij u and its inverse gij u are assumed to satisfy the hyperbolicity condition that is have signature n 1 . The functions gij gij and qij are assumed to be smooth bounded and have globally bounded derivatives as functions of u. To insure that the level surfaces of t are space-like we assume that g00 1. We then consider the following question For which values of s is the problem 1.1 and 1.2 locally well-posed in Hs X Hs-1 In general well-posedness involves existence uniqueness and continuous dependence on the initial data. Naively one would hope to have these properties hold for solutions in C Hs n C 1 Hs-1 but it appears that there is little chance to establish uniqueness under this condition for the low values of s that we consider in this paper. Our definition of well-posedness thus includes The research of the first author was partially supported by NSF grant DMS-9970407. The research of the second author was partially supported by NSF grant DMS-9970297. 292 HART F. SMITH AND DANIEL TATARU an additional assumption on the solution u to insure uniqueness while also providing useful information about the solution. Definition 1.1. We say that the Cauchy problem 1.1 .