tailieunhanh - Đề tài " Sharp local wellposedness results for the nonlinear wave equation "

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 . The results. We consider in this paper second order, nonlinear hyperbolic equations of the form () gij (u) ∂i ∂j u = q ij (u) ∂i u ∂j u on R × Rn , with Cauchy data prescribed at time 0, () 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 . The results. We consider in this paper second order nonlinear hyperbolic equations of the form gij u didju qij u diudju on R X Rra with Cauchy data prescribed at time 0 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 and 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 . We say that the Cauchy problem .