On the quasilinear wave equations in time dependent inhomogeneous media

Abstract

We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds (R 3+1 , g) with time dependent inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely C 1 close to the Minkowski metric inside some large cylinder { (t, x)| |x| ≤ R} and approaches the Minkowski metric weakly as |x| → ∞. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equations such that the solution is globally stable under perturbations of initial data. * This work is part of the author's Ph.D. thesis at Princeton University. Preliminaries and Energy MethodGiven any Lorentzian metric g = g µν dx µ dx ν , x 0 = t on R 3+1 , we let g µν denotes the components of the inverse of the metric g. Throughout this paper, we let A, B be any vector fields in {L, L, S 1 , S 2 } and S be any vector fields in {S 1 , S 2 }. Relative to the null frame, the metric components are g AB . The inverse is g AB . We denote ∂ µ = g µν ∂ ν , ∂ A = g AB B.

Global stability of solutions to nonlinear wave equations

“…The corresponding problem for the general quasilinear wave equations will be discussed in the author's forthcoming paper [31]. However, in that paper, the assumption on ∂ is stronger which is of the form (4).…”

Some remarks on quasilinear wave equations with null condition in 3‐D

“…Moreover, in Alinhac [3], it is essentially assumed that the initial data have a compact support. However, the Lorentz boost operator is not suitable for some wave systems, which are not Lorentz invariant, such as nonrealistic system with multiple wave speeds [7][8][9], nonlinear wave equations on non-flat space-time [10][11][12][13], wave-type equation with nonlocal term [14,15] and exterior problem [16]. Furthermore, the compact support assumption of the initial data is also restrictive and is not suitable for some nonlocal problems such as the incompressible elastodynamics(e.g.…”

A vector field method for non-trapping, radiating spacetimes

“…Since the appearance of that work there has been a concerted effort to extend the vector field method to backgrounds far from Minkowski space. In many cases these efforts have been successful and have yielded small data global existence results for non-linear problems in spacetimes such as: Minkowski space with obstacles [26], exterior Kerr with |a| ≪ M ( [20], [24]) and time-dependent, inhomogeneous media ( [39], [38])…”