Nonlinear stability of blow-up solutions to the Hyperbolic Mean Curvature flow

[摘要] We study the so-called Hyperbolic Mean Curvature (HMC) flow introduced by LeFloch and Smoczyk in 2008 for the evolution of a closed hypersurface moving in the direction of its mean curvature vector. This flow stems from a geometrically natural action consisting of a kinetic energy and an internal energy. We study the initial value problem for this flow in the case of an entire graph (in arbitrary dimension) and we establish the existence of a (singular) self-similar solution and its nonlinear stability in a suitably weighted Sobolev space by relying on Nash-Moser iterations. (c) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-05

[关键词] Hyperbolic Mean Curvature flow;Self-similar solution;Quasilinear wave equation;Nonlinear stability;Weighted Sobolev space