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.
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.
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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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