Abstract 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.

中文翻译：

---

双曲平均曲率流爆破解的非线性稳定性

摘要 我们研究了 LeFloch 和 Smoczyk 在 2008 年引入的所谓双曲平均曲率 (HMC) 流，用于沿其平均曲率向量方向移动的闭合超曲面的演化。这种流动源于由动能和内能组成的几何自然作用。我们在整个图（任意维度）的情况下研究了该流的初始值问题，并且我们通过依赖 Nash 建立了（奇异的）自相似解及其在适当加权的 Sobolev 空间中的非线性稳定性 -莫泽迭代。