Hyperbolic Mean Curvature Research Articles

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

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.

Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.