Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity

Hughes, T. J. R., T. Kato and J. E. Marsden

Arch. Rational Mech. Anal., 63, (1977), 273-294

Abstract:

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 Kato [29], are applied to second-order quasi-linear hyperbolic systems on $\mathbb {R}$ⁿ whose solutions (u(t) ,$\dot{{u}}$(t)) lie in the Sobolev space H^s+1 x H^s . 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 > 1.5; s $\geq$ 3 was the best known previous value for systems of the type occuring in general relativity.