Strongly interacting blow up bubbles  for the mass critical nonlinear Schrödinger equation

Martel, Yvan; Raphaël, Pierre

doi:10.24033/asens.2364

Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation
[Bulles d'explosion en interaction forte pour l'équation de Schrödinger non linéaire critique pour la masse]

Martel, Yvan ; Raphaël, Pierre

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 701-737.

Résumé
Abstract

On considère l'équation de Schrödinger non linéaire critique pour la masse en dimension deux

i \partial_{t} u + Δ u + {| u |}^{2} u = 0, t \in ℝ, x \in ℝ^{2} . (SNL)

Soit

Q

la solution positive et état fondamental de l'équation

Δ Q - Q + Q^{3} = 0

. On construit une nouvelle classe d'ondes solitaires multiples basées sur

Q

: étant donné un entier

K \geq 2

, il existe une solution globale (pour

t > 0

)

u (t)

de (SNL) qui se décompose asymptotiquement en une somme d'ondes solitaires centrées sur les sommets d'un polygone régulier et qui se concentrent à un taux logarithmique quand

t \to + \infty

, de sorte que la solution explose en temps infini

{∥ \nabla u (t) ∥}_{L^{2}} \sim | log t | quand t \to + \infty .

Comme conséquence de la symétrie pseudo-conforme du flot de (SNL), on obtient le premier exemple d'une solution

v (t)

de (SNL) qui explose en temps fini avec un taux strictement supérieur au taux pseudo-conforme

{∥ \nabla v (t) ∥}_{L^{2}} \sim |\frac{log | t |}{t}| quand t ↑ 0 .

Cette solution concentre

K

bulles en un point

x_{0} \in ℝ^{2}

, c'est-à-dire

{| v (t) |}^{2} ⇀ K {∥ Q ∥}_{L^{2}}^{2} δ_{x_{0}}

quand

t ↑ 0 .

Ces comportements particuliers sont dus aux interactions fortes entre les ondes solitaires, par opposition avec les résultats précédents sur les ondes solitaires multiples pour (SNL) où les interactions n'affectent pas le comportement global des ondes.

We consider the mass critical two dimensional nonlinear Schrödinger equation

i \partial_{t} u + Δ u + {| u |}^{2} u = 0, t \in ℝ, x \in ℝ^{2} . (NLS)

Let

Q

denote the positive ground state solution of

Δ Q - Q + Q^{3} = 0

. We construct a new class of multi-solitary wave solutions of (NLS) based on

Q

: given any integer

K \geq 2

, there exists a global (for

t > 0

) solution

u (t)

that decomposes asymptotically into a sum of solitary waves centered at the vertices of a

K

-sided regular polygon and concentrating at a logarithmic rate as

t \to + \infty

, so that the solution blows up in infinite time with the rate

{∥ \nabla u (t) ∥}_{L^{2}} \sim | log t | as t \to + \infty .

Using the pseudo-conformal symmetry of the (NLS) flow, this yields the first example of solution

v (t)

of (NLS) blowing up in finite time with a rate strictly above the pseudo-conformal one, namely,

{∥ \nabla v (t) ∥}_{L^{2}} \sim |\frac{log | t |}{t}| as t ↑ 0 .

Such a solution concentrates

K

bubbles at a point

x_{0} \in ℝ^{2}

, that is

{| v (t) |}^{2} ⇀ K {∥ Q ∥}_{L^{2}}^{2} δ_{x_{0}}

t ↑ 0 .

These special behaviors are due to strong interactions between the waves, in contrast with previous works on multi-solitary waves of (NLS) where interactions do not affect the global behavior of the waves.

Publié le : 2018-11-12

DOI : 10.24033/asens.2364

Classification : 35Q55; 35B44, 37K40.
Keywords: Nonlinear Schrödinger equation, critical nonlinearity, blow up, multi-solitons
Mot clés : Équation de Schrödinger non linéaire, non-linéarité critique, explosion, multi-solitons.

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     author = {Martel, Yvan and Rapha\"el, Pierre},
     title = {Strongly interacting blow up bubbles  for the mass critical nonlinear {Schr\"odinger} equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {701--737},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {3},
     year = {2018},
     doi = {10.24033/asens.2364},
     mrnumber = {3831035},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2364/}
}

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Martel, Yvan; Raphaël, Pierre. Strongly interacting blow up bubbles  for the mass critical nonlinear Schrödinger equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 701-737. doi : 10.24033/asens.2364. http://www.numdam.org/articles/10.24033/asens.2364/

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