Validity of nonlinear geometric optics with times growing logarithmically

Abstract:

The profiles (a.k.a. amplitudes) which enter in the approximate solutions of nonlinear geometric optics satisfy equations, sometimes called the slowly varying amplitude equations, which are simpler than the original hyperbolic systems. When the underlying problem is conservative one often finds that the amplitudes are defined for all time and are uniformly bounded. The approximations of nonlinear geometric optics typically have percentage error which tends to zero uniformly on bounded time intervals as the wavelength $\epsilon$ tends to zero. Under suitable hypotheses when the amplitude is uniformly bounded in space and time we show that the percentage error tends to zero uniformly on time intervals $[0,C|\ln \epsilon |]$ which grow logarithmically. The proof relies in an essential way on the fact that one has a corrector to the leading term of geometric optics.