On almost global existence for nonrelativistic wave equations in 3D

Abstract

Almost global solutions are constructed to three‐dimensional, quadratically nonlinear wave equations. The proof relies on generalized energy estimates and a new decay estimate. The method applies to equations that are only classically invariant, such as the nonlinear system of hyperelasticity. © 1996 John Wiley & Sons, Inc.

A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations

“…In compressible nonviscous fluid mechanics, the field ∂ t + u.∇ x essentially governs the behavior of the vortex, and again, this prevents using the h i . In [20], Klainerman and Sideris have found substitutes for Klainerman's inequality in these cases. These ideas have also been used by Sideris to study nonlinear elasticity [23].…”

Free decay of solutions to wave equations on a curved background

“…If we do not use the H i , we have to rely on the substitute formula found by Klainerman and Sideris [10]. These formula allow roughly to control σ∂ 2 v by ∂Sv, ∂Rv, ∂v and (1 + t + r)L 0 v. Here, we have to adapt them to allow a control by L g , and this is the reason why we require a special behavior of the lower order terms coefficientsθ and k in L g , and why we make our technical interior assumption.…”

“…Klainerman and Sideris [10] have established a substitute for i) in the flat case. In the following proposition, we establish similar but more geometric formula involving L g .…”

“…c) We follow now the proof of [10], trying to replace ∂ t and ∂ r by T and N whenever possible. We thus write…”

“…A more precise discussion of these issues will be offered in section 1.4 after our notations, assumptions and results have been stated. Let us just say here that the whole paper is strongly inspired by the geometric techniques of Christodoulou and Klainerman, developed in [4], [3] and also by related work of Klainerman and Sideris [10], Klainerman and Nicolò [8] and Klainerman and Rodnianski [9].…”