[Sur la formation de singularités pour le flot hyperbolique de courbure moyenne nulle de surfaces asymptotiques au cône de Simons]
Dans cette note, on étabit l'existence d'une famille de surfaces 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 , où Q est une solution stationnaire et 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 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 , 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.
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Hajer Bahouri 1 ; Alaa Marachli 1 ; Galina Perelman 1
@article{CRMATH_2019__357_10_778_0, author = {Hajer Bahouri and Alaa Marachli and Galina Perelman}, title = {Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a {Simons} cone}, journal = {Comptes Rendus. Math\'ematique}, pages = {778--783}, publisher = {Elsevier}, volume = {357}, number = {10}, year = {2019}, doi = {10.1016/j.crma.2019.10.001}, language = {en}, }
TY - JOUR AU - Hajer Bahouri AU - Alaa Marachli AU - Galina Perelman TI - Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone JO - Comptes Rendus. Mathématique PY - 2019 SP - 778 EP - 783 VL - 357 IS - 10 PB - Elsevier DO - 10.1016/j.crma.2019.10.001 LA - en ID - CRMATH_2019__357_10_778_0 ER -
%0 Journal Article %A Hajer Bahouri %A Alaa Marachli %A Galina Perelman %T Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone %J Comptes Rendus. Mathématique %D 2019 %P 778-783 %V 357 %N 10 %I Elsevier %R 10.1016/j.crma.2019.10.001 %G en %F CRMATH_2019__357_10_778_0
Hajer Bahouri; Alaa Marachli; Galina Perelman. Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 778-783. doi : 10.1016/j.crma.2019.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.001/
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