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Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

IntroductionIn this paper we study global behavior of solution of second-order nonlinear hyperbolic equations: It is a well-known fact that in one dimension solutions of this type of equations tend to develop singularities after a finite time, no matter how smooth and "small" the initial data one starts with are (see and H E C;(W), develops a shock at t = -l/h, x = 6, where h = min H" = H"(6).More precisely, u,, = m at this point, while u, u,, u, are still bounded (see [l]).Therefore u, as a classical solution of (l), (2), has a "life span" T = -l/h. More generally, one can provide lower bounds for the "life span" of solutions of second-order hyperbolic equations We suppose G is a smooth function of h' =((Ai), i = O ; a , n. Writing u ( l ) = ef, u(')= eg we have 1/T= O ( E ) for E+O.It has been shown recently by Fritz John [l] that for n > 1 these lower bounds for T can be improved. More precisely, for quasilinear equations of the form where a,, are smooth in a neighborhood of the origin, and aii(0) at t = 0 is a positive definite symmetric matrix,for n > 3 GLOBAL EXISTENCE FOR NONLINEAR WAVE EQUATIONS 45This long-time existence for solutions of (3) (1 1)The following natural question arises. Can one actually use (12) to show that smooth solutions of (3) exist for all time if the initial data are small enough. The aim of this paper is to show that the answer is yes, at least for space dimensions greater than or equal to 6 .Before stating the result we describe the following example suggested to me by L. Nirenberg.Consider the semilinear hyperbolic equation Making the transformation u = e' we transform (13) into This last equation can be solved explicitly and we obtain To state our result we have to put a slight restriction on the linear part of (3). In fact we shall assume, modulo a linear change of coordinates, that G has the form For the purpose of this introduction we choose a particular form for F. We can assume that F is a quadratic. As a matter of fact higher-order terms are much easier to handle. This is reflected in the following:Remark. If F(X) = O(la13) near = 0 and n 2 4 , a global smooth solution of (3) exist for small initial data. Moreover, this can be proved using a classical Picard iteration scheme. The case n = 3 will be treated separately in a future paper.For the remainder of this introduction we shall discuss Theorem 2 in the following situation: is easily derived by integration by parts. Also, being a little more careful (see Section 4), one can show thatTo keep I l~( t ) I l~.~ bounded for all t ZO, we need decay in s for IDu(s)lL-.l at least like O(S-'-').Therefore, in order to control Ilt+,llE.N in the iteration (24), we need to have decay estimates in sup norm for t+,+l. The main difficulty consists now in showing that up has the same decay properties asIn other words, we are forced to derive a priori estimates for the supnorm of the solution u of (26). To do this we write (26) in the form W(t-s) h(s;)ds,where W(t')h is the operator solution of the homogeneous wave equation.Taking der...

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Library of Congress Cataloging•in•Publication Data Klainerman, Sergiu, 1950 The evolution problem in general relativity / Sergiu Klainerman and Francesco Nicolo. p. cm.-(Progress in mathematical physics; v. 25) Includes bibliographical references and index.