Blowup for biharmonic NLS

Boulenger, Thomas; Lenzmann, Enno

doi:10.24033/asens.2326

Blowup for biharmonic NLS
[Phénomènes d'explosion pour NLS Biharmonique]

Boulenger, Thomas ; Lenzmann, Enno

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 503-544.

Résumé
Abstract

On considère le problème de Cauchy pour NLS biharmonique (i.e., d'ordre quatre) focalisante définie par

i \partial_{t} u = Δ^{2} u - μ Δ u - {| u |}^{2 σ} u for (t, x) \in [0, T) \times ℝ^{d},

avec

0 < σ < \infty

pour

d \leq 4

0 < σ \leq 4 / (d - 4)

pour

d \geq 5

; et

μ \in ℝ

est un paramètre destiné à éventuellement inclure un terme dispersif d'ordre inférieur. Dans le cas sur-critique

σ > 4 / d

, on prouve un résultat général d'explosion en temps fini pour des données radiales dans

H^{2} (ℝ^{d})

en toute dimension

d \geq 2

. On déduit par ailleurs une borne supérieure universelle pour la vitesse d'explosion moyennée en temps pour certains indices

4 / d < σ < 4 / (d - 4)

. Dans le cas critique

σ = 4 / d

, on prouve ensuite un résultat général d'explosion en temps fini ou infini, toujours pour des solutions à données radiales

H^{2} (ℝ^{d})

. On utilise là de façon cruciale l'évolution temporelle d'une quantité positive, que nous baptisons la bivariance (locale) de Riesz pour NLS biharmonique. Cette quantité nous sert de substitut avantageux à la variance classiquement utilisée pour l'étude des problèmes NLS.

On prouve enfin l'existence d'un ground state radial pour NLS biharmonique, qui pourra s'avérer utile pour l'étude du problème elliptique associé.

We consider the Cauchy problem for the biharmonic (i.e., fourth-order) NLS with focusing nonlinearity given by

i \partial_{t} u = Δ^{2} u - μ Δ u - {| u |}^{2 σ} u for (t, x) \in [0, T) \times ℝ^{d},

where

0 < σ < \infty

for

d \leq 4

and

0 < σ \leq 4 / (d - 4)

for

d \geq 5

; and

μ \in ℝ

is some parameter to include a possible lower-order dispersion. In the mass-supercritical case

σ > 4 / d

, we prove a general result on finite-time blowup for radial data in

H^{2} (ℝ^{d})

in any dimension

d \geq 2

. Moreover, we derive a universal upper bound for the blowup rate for suitable

4 / d < σ < 4 / (d - 4)

. In the mass-critical case

σ = 4 / d

, we prove a general blowup result in finite or infinite time for radial data in

H^{2} (ℝ^{d})

. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems.

In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.

Publié le : 2017-11-12

DOI : 10.24033/asens.2326

Classification : 35Q55, 35J48, 35A01, 31B30.
Keywords: Biharmonic NLS, fourth-order NLS, blowup, ground states.
Mot clés : NLS biharmonique, NLS d'ordre quatre, phénomènes d'explosion, ground states.

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     author = {Boulenger, Thomas and Lenzmann, Enno},
     title = {Blowup for biharmonic {NLS}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {503--544},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {3},
     year = {2017},
     doi = {10.24033/asens.2326},
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Boulenger, Thomas; Lenzmann, Enno. Blowup for biharmonic NLS. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 503-544. doi : 10.24033/asens.2326. http://www.numdam.org/articles/10.24033/asens.2326/

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