Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone

Bahouri, Hajer; Marachli, Alaa; Perelman, Galina

doi:10.1016/j.crma.2019.10.001

Partial differential equations

Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone
[Sur la formation de singularités pour le flot hyperbolique de courbure moyenne nulle de surfaces asymptotiques au cône de Simons]

Présenté par : the Editorial Board

Bahouri, Hajer ¹ ; Marachli, Alaa ¹ ; Perelman, Galina ¹

¹ Université Paris-Est Créteil, UMR 8050, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France

Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 778-783.

Résumé
Abstract

Dans cette note, on étabit l'existence d'une famille de surfaces ${(Γ (t))}_{0 < t \leq T}$ qui évoluent sous le flot de courbure moyenne nulle dans l'espace de Minkowski et qui explosent lorsque t tend vers 0 vers une surface asymptotique au cône de Simons à l'infini. Ce problème revient à étudier la formation de singularités pour une équation d'ondes quasi-linéaire du second ordre. Notre approche constructive consiste à démontrer l'existence de solutions à cette équation hyperbolique explosant en temps fini sous la forme $u (t, x) \sim t^{ν + 1} Q (\frac{x}{t^{ν + 1}})$ , où Q est une solution stationnaire et $ν > 1 / 2$ est un nombre irrationnel. Notre démarche s'inspire de celle de Krieger, Schlag et Tataru dans [7–9]. Cependant contrairement à ces travaux, l'équation en question dans cette note est quasi-linéaire, ce qui génère des difficultés que l'on doit surmonter.

In this paper, we establish the existence of a family of surfaces ${(Γ (t))}_{0 < t \leq T}$ that evolve by the vanishing mean curvature flow in Minkowski space and, as t tends to 0, blow up towards a surface that behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second-order quasilinear wave equation. Our constructive approach consists in proving the existence of a finite-time blow-up solution to this hyperbolic equation under the form $u (t, x) \sim t^{ν + 1} Q (\frac{x}{t^{ν + 1}})$ , where Q is a stationary solution and ν is an irrational number strictly larger than 1/2. Our strategy roughly follows that of Krieger, Schlag and Tataru in [7–9]. However, contrary to these articles, the equation to be handled in this work is quasilinear. This induces a number of difficulties to face.

Reçu le : 2019-01-27
Accepté le : 2019-10-07
Publié le : 2019-10-01

DOI : 10.1016/j.crma.2019.10.001

Affiliations des auteurs :

Bahouri, Hajer ¹ ; Marachli, Alaa ¹ ; Perelman, Galina ¹

¹ Université Paris-Est Créteil, UMR 8050, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France

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     title = {Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a {Simons} cone},
     journal = {Comptes Rendus. Math\'ematique},
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Bahouri, Hajer; Marachli, Alaa; Perelman, Galina. Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone. Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 778-783. doi : 10.1016/j.crma.2019.10.001. http://www.numdam.org/articles/10.1016/j.crma.2019.10.001/

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