Abstract
We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.
Highlights
The answer to the question asked in the previous section is in the affirmative: we show that simple planewave solutions to the membrane equation are stable under small initial data perturbations
The nonlinearities do not introduce new difficulties beyond the adjustments made for the modified linear evolution. Another difference with our work and [DKSW16] is that they prove that the catenoid is globally stable under axially symmetric codimension one initial perturbations, whereas we prove that our plane-wave solution is globally stable under an open set of symmetrybreaking perturbations
To help manage the nonlinearities that arise in such arguments in a systematic way, we introduce in this paper a weighted vector field algebra
Summary
This is a special geometric feature of simple travelling-wave solutions to the membrane equation To expose this special linear structure, one needs to make an appropriate gauge choice involving a nonlinear change of variables adapted to the background Υ , which essentially rewrites our perturbation equations as a graph in the normal bundle of Υ , interpreted as a submanifold of R1,1+d. Putting this together with the fact that the nonlinearities in (1.1) are cubic, this means that heuristically we can understand the result of [WW17] as very similar to the large-data stability result for the wave map system proven in [Sid89], which required the ‘background geodesic solution’ to be one with finite (weighted) energy, and decays like finite energy solutions to the linear wave equation These types of systems can be modelled by the quasilinear system. We include in the Appendix B a list of notations that are introduced and references to their definitions
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