Geodesic
Blowup for biharmonic NLS
[Phénomènes d'explosion pour NLS Biharmonique]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 503-544.

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

i∂tu=Δ2u-μΔu-|u|2σufor(t,x)∈[0,T)×ℝd,
where 0<σ<∞ for d≤4 and 0<σ≤4/(d-4) for d≥5; and μ∈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 H2(ℝd) in any dimension d≥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 H2(ℝd). 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.

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

i∂tu=Δ2u-μΔu-|u|2σufor(t,x)∈[0,T)×ℝd,
avec 0<σ<∞ pour d≤4 et 0<σ≤4/(d-4) pour d≥5; et μ∈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 H2(ℝd) en toute dimension d≥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 H2(ℝd). 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.

Publié le :
DOI : 10.24033/asens.2326
Classification : 35Q55, 35J48, 35A01, 31B30.
Keywords: Biharmonic NLS, fourth-order NLS, blowup, ground states.
Mots-clés : NLS biharmonique, NLS d'ordre quatre, phénomènes d'explosion, ground states. 
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     title = {Blowup for biharmonic {NLS}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {503--544},
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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://geodesic.mathdoc.fr/articles/10.24033/asens.2326/

Ben-Artzi, M.; Koch, H.; Saut, J.-C. Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000), pp. 87-92 (ISSN: 0764-4442) | MR | Zbl | DOI

Baruch, G.; Fibich, G. Singular solutions of the L2-supercritical biharmonic nonlinear Schrödinger equation, Nonlinearity, Volume 24 (2011), pp. 1843-1859 (ISSN: 0951-7715) | MR | Zbl | DOI

Baruch, G.; Fibich, G.; Mandelbaum, E. Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation, Nonlinearity, Volume 23 (2010), pp. 2867-2887 (ISSN: 0951-7715) | MR | Zbl | DOI

Baruch, G.; Fibich, G.; Mandelbaum, E. Singular solutions of the biharmonic nonlinear Schrödinger equation, SIAM J. Appl. Math., Volume 70 (2010), pp. 3319-3341 (ISSN: 0036-1399) | MR | Zbl | DOI

Bellazzini, J.; Frank, R. L.; Visciglia, N. Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems, Math. Ann., Volume 360 (2014), pp. 653-673 (ISSN: 0025-5831) | MR | Zbl | DOI

Boulenger, T.; Himmelsbach, D.; Lenzmann, E. Blowup for fractional NLS, J. Funct. Anal., Volume 271 (2016), pp. 2569-2603 (ISSN: 0022-1236) | DOI | MR

Boulenger, T.; Lenzmann, E. (work in preparation)

Brascamp, H. J.; Lieb, E. H.; Luttinger, J. M. A general rearrangement inequality for multiple integrals, J. Functional Analysis, Volume 17 (1974), pp. 227-237 | MR | Zbl | DOI

Cazenave, T., Courant Lecture Notes in Math., 10, New York University, Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 2003, 323 pages (ISBN: 0-8218-3399-5) | MR | Zbl | DOI

Carlen, E. A.; Loss, M. Extremals of functionals with competing symmetries, J. Funct. Anal., Volume 88 (1990), pp. 437-456 (ISSN: 0022-1236) | MR | Zbl | DOI

Chen, W.; Li, C.; Ou, B. Classification of solutions for an integral equation, Comm. Pure Appl. Math., Volume 59 (2006), pp. 330-343 (ISSN: 0010-3640) | MR | Zbl | DOI

Cho, Y.; Ozawa, T.; Wang, C. Finite time blowup for the fourth-order NLS, Bull. Korean Math. Soc., Volume 53 (2016), pp. 615-640 (ISSN: 1015-8634) | DOI | MR

Duyckaerts, T.; Holmer, J.; Roudenko, S. Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., Volume 15 (2008), pp. 1233-1250 (ISSN: 1073-2780) | MR | Zbl | DOI

Duoandikoetxea, J. Fractional integrals on radial functions with applications to weighted inequalities, Ann. Mat. Pura Appl., Volume 192 (2013), pp. 553-568 (ISSN: 0373-3114) | MR | Zbl | DOI

Fibich, G.; Ilan, B.; Papanicolaou, G. Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., Volume 62 (2002), pp. 1437-1462 (ISSN: 0036-1399) | MR | Zbl | DOI

Fröhlich, J.; Lenzmann, E. Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., Volume 60 (2007), pp. 1691-1705 (ISSN: 0010-3640) | MR | Zbl | DOI

Guevara, C. D. Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, Volume 2 (2014), pp. 177-243 (ISSN: 1687-1200) | MR | Zbl

Ginibre, J.; Velo, G. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., Volume 32 (1979), pp. 1-32 (ISSN: 0022-1236) | MR | Zbl | DOI

Holmer, J.; Roudenko, S. On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, Volume 1 (2007) (ISSN: 1687-1200) | MR | Zbl

Holmer, J.; Roudenko, S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., Volume 282 (2008), pp. 435-467 (ISSN: 0010-3616) | MR | Zbl | DOI

Karpman, V. I. Stabilization of soliton instabilities by higher order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Lett. E, Volume 53 (1996), pp. 1336-1339 (ISSN: 0375-9601) | DOI | MR

Kenig, C. E.; Merle, F. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. math., Volume 166 (2006), pp. 645-675 (ISSN: 0020-9910) | MR | Zbl | DOI

Karpman, V. I.; Shagalov, A. G. Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, Volume 144 (2000), pp. 194-210 (ISSN: 0167-2789) | MR | Zbl | DOI

Killip, R.; Visan, M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., Volume 132 (2010), pp. 361-424 (ISSN: 0002-9327) | MR | Zbl | DOI

Li, Y. Y. Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS), Volume 6 (2004), pp. 153-180 http://link.springer.de/... (ISSN: 1435-9855) | MR | Zbl | DOI

Lieb, E. H. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., Volume 118 (1983), pp. 349-374 (ISSN: 0003-486X) | MR | Zbl | DOI

Lieb, E. H.; Loss, M., Graduate Studies in Math., 14, Amer. Math. Soc., Providence, RI, 2001, 346 pages (ISBN: 0-8218-2783-9) | MR | Zbl | DOI

Merle, F.; Raphaël, P.; Szeftel, J. On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation, Duke Math. J., Volume 163 (2014), pp. 369-431 (ISSN: 0012-7094) | MR | Zbl | DOI

Merle, F.; Tsutsumi, Y. L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, Volume 84 (1990), pp. 205-214 (ISSN: 0022-0396) | MR | Zbl | DOI

Miao, C.; Xu, G.; Zhao, L. Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, Volume 246 (2009), pp. 3715-3749 (ISSN: 0022-0396) | MR | Zbl | DOI

Ogawa, T.; Tsutsumi, Y. Blow-up of H1 solution for the nonlinear Schrödinger equation, J. Differential Equations, Volume 92 (1991), pp. 317-330 (ISSN: 0022-0396) | MR | Zbl | DOI

Pausader, B. Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., Volume 4 (2007), pp. 197-225 (ISSN: 1548-159X) | MR | Zbl | DOI

Pausader, B. The cubic fourth-order Schrödinger equation, J. Funct. Anal., Volume 256 (2009), pp. 2473-2517 (ISSN: 0022-1236) | MR | Zbl | DOI

Pausader, B. The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., Volume 24 (2009), pp. 1275-1292 (ISSN: 1078-0947) | MR | Zbl | DOI

Pausader, B.; Shao, S. The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyperbolic Differ. Equ., Volume 7 (2010), pp. 651-705 (ISSN: 0219-8916) | MR | Zbl | DOI

Pausader, B.; Xia, S. Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, Volume 26 (2013), pp. 2175-2191 (ISSN: 0951-7715) | MR | Zbl | DOI

Reed, M.; Simon, B., Academic Press, New York-London, 1975, 361 pages | MR | Zbl

Sulem, C.; Sulem, P.-L., Applied Mathematical Sciences, 139, Springer, New York, 1999, 350 pages (ISBN: 0-387-98611-1) | MR | Zbl

Strauss, W. A. Existence of solitary waves in higher dimensions, Comm. Math. Phys., Volume 55 (1977), pp. 149-162 http://projecteuclid.org/euclid.cmp/1103900983 (ISSN: 0010-3616) | MR | Zbl | DOI

Weinstein, M. I., The connection between infinite-dimensional and finite-dimensional dynamical systems (Boulder, CO, 1987) (Contemp. Math.), Volume 99, Amer. Math. Soc., Providence, RI, 1989, pp. 213-232 | MR | Zbl | DOI

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