2019/02/19 by Hajer Bahouri, Bahouri, Hajer, Alaa Marachli +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1902.07027
openalex publication_date 2019/02/19 · openalex created_date 2023/01/02 · openalex updated_date 2026/07/28
In this article, we establish the existence of a family of hypersurfaces\n(\Γ (t))0< t \≤ T which evolve by the vanishing mean curvature flow\nin Minkowski space and which as t tends to~0 blow up towards a hypersurface\nwhich behaves like the Simons cone at infinity. This issue amounts to\ninvestigate the singularity formation for a second order quasilinear wave\nequation. Our constructive approach consists in proving the existence of finite\ntime blow up solutions of this hyperbolic equation under the form u(t,x) \∼\nt^ \ν+1 Q\( frac x t^ \ν+1 \) , where~Q is a stationary\nsolution and \ν an arbitrary large positive irrational number. Our approach\nroughly follows that of Krieger, Schlag and Tataru. However contrary to these\nworks, the equation to be handled in this article is quasilinear. This induces\na number of difficulties to face.\n