Hughes , Thomas J. R. and Kato, Tosio and Marsden, Jerrold E. (1977) Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Archive for Rational Mechanics and Analysis, 63 (3). pp. 273-294. ISSN 0003-9527 http://resolver.caltech.edu/CaltechAUTHORS:20100817-114853033
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Existence, uniqueness and well-posedness for a general class of quasi-linear evolution equations on a short time interval are established. These results, generalizing those of , are applied to second-order quasi-linear hyperbolic systems on IR" whose solutions (u(t),il(t)) lie in the Sobolev space H^(s+1) x H^5. Our results improve existing theorems by lowering the required value of s to s > (n/2) + 1, or s > n/2 in case the coefficients of the highest order terms do not involve derivatives of the unknown, and by establishing continuous dependence on the initial data for these values. As consequences we obtain well-posedness of the equations of elastodynamics if s>2.5 and of general relativity if s>l.5; s>3 was the best known previous value for systems of the type occuring in general relativity (, , ).
|Additional Information:||© 1977. Received March 25, 1976. We thank S. ANTMAN, Y. CHOQUET-BRUHAT and A. FISCHER for several useful remarks. This work was partially supported by NSF grants GP-37780X and MPS-75-05576. Communicated by S. ANTMAN|
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|Deposited By:||Ruth Sustaita|
|Deposited On:||17 Aug 2010 20:42|
|Last Modified:||26 Dec 2012 12:19|
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