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

Martel, Yvan; Raphaël, Pierre

doi:10.24033/asens.2364

Geodesic

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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.

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Abstract (VO)
Résumé

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.

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.

Publié le : 2018-05-27

DOI : 10.24033/asens.2364

Classification : 35Q55; 35B44, 37K40.
Keywords: Nonlinear Schrödinger equation, critical nonlinearity, blow up, multi-solitons
Mots-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},
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     mrnumber = {3831035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.24033/asens.2364/

Bibliographie
Cité par

Antonini, C. Lower bounds for the $L^{2}$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Differential Integral Equations, Volume 15 (2002), pp. 749-768 (ISSN: 0893-4983) | MR | Zbl | DOI

Banica, V. Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 3 (2004), pp. 139-170 (ISSN: 0391-173X) | MR | Zbl | mathdoc-id

Banica, V.; Carles, R.; Duyckaerts, T. Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation, Comm. Partial Differential Equations, Volume 36 (2011), pp. 487-531 (ISSN: 0360-5302) | MR | Zbl | DOI

Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 313-345 (ISSN: 0003-9527) | MR | Zbl | DOI

Boulenger, T. Minimal mass blow up for NLS on a manifold (2012)

Bourgain, J.; Wang, W. Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 25 (1997), pp. 197-215 (ISSN: 0391-173X) | MR | Zbl | mathdoc-id

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

Chang, S.-M.; Gustafson, S.; Nakanishi, K.; Tsai, T.-P. Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., Volume 39 (2007/08), pp. 1070-1111 (ISSN: 0036-1410) | MR | Zbl | DOI

Côte, R.; Martel, Y.; Merle, F. Construction of multi-soliton solutions for the $L^{2}$ -supercritical generalized Korteweg-de Vries and NLS equations, Rev. Mat. Iberoam., Volume 27 (2011), pp. 273-302 | MR | Zbl | DOI

Collot, C. Type II blow up manifold for the energy super critical wave equation (preprint arXiv:1407.4525, to appear in Mem. Amer. Math. Soc ) | MR

Cazenave, T.; Weissler, F. B. The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$ , Nonlinear Anal., Volume 14 (1990), pp. 807-836 (ISSN: 0362-546X) | MR | Zbl | DOI

Donninger, R.; Huang, M.; Krieger, J.; Schlag, W. Exotic blowup solutions for the $u^{5}$ focusing wave equation in $ℝ^{3}$ , Michigan Math. J., Volume 63 (2014), pp. 451-501 (ISSN: 0026-2285) | MR | Zbl | DOI

Duyckaerts, T.; Kenig, C. E.; Merle, F. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations, Commun. Pure Appl. Anal., Volume 14 (2015), pp. 1275-1326 (ISSN: 1534-0392) | DOI | MR

Duyckaerts, T.; Merle, F. Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., Volume 18 (2009), pp. 1787-1840 (ISSN: 1016-443X) | MR | Zbl | DOI

Dodson, B. Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., Volume 285 (2015), pp. 1589-1618 (ISSN: 0001-8708) | DOI | MR

Fan, C. Log-log blow up solutions blow up at exactly $m$ points (preprint arXiv:1510.00961 ) | MR | mathdoc-id

Herrero, M. A.; Velázquez, J. J. L. Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, Volume 5 (1992), pp. 973-997 (ISSN: 0893-4983) | MR | Zbl | DOI

Jendrej, J. Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5, J. Funct. Anal., Volume 272 (2017), pp. 866-917 (ISSN: 0022-1236) | DOI | MR

Krieger, J.; Lenzmann, E.; Raphaël, P. Nondispersive solutions to the $L^{2}$ -critical half-wave equation, Arch. Ration. Mech. Anal., Volume 209 (2013), pp. 61-129 (ISSN: 0003-9527) | MR | Zbl | DOI

Krieger, J.; Martel, Y.; Raphaël, P. Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math., Volume 62 (2009), pp. 1501-1550 (ISSN: 0010-3640) | MR | Zbl | DOI

Krieger, J.; Schlag, W. Non-generic blow-up solutions for the critical focusing NLS in 1-D, J. Eur. Math. Soc., Volume 11 (2009), pp. 1-125 | MR | Zbl | DOI

Krieger, J.; Schlag, W. Full range of blow up exponents for the quintic wave equation in three dimensions, J. Math. Pures Appl., Volume 101 (2014), pp. 873-900 (ISSN: 0021-7824) | MR | Zbl | DOI

Krieger, J.; Schlag, W.; Tataru, D. Renormalization and blow up for charge one equivariant critical wave maps, Invent. math., Volume 171 (2008), pp. 543-615 (ISSN: 0020-9910) | MR | Zbl | DOI

Killip, R.; Tao, T.; Visan, M. The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), Volume 11 (2009), pp. 1203-1258 (ISSN: 1435-9855) | MR | Zbl | DOI

Kwong, M. K. Uniqueness of positive solutions of $Δ u - u + u^{p} = 0$ in $𝐑^{n}$ , Arch. Rational Mech. Anal., Volume 105 (1989), pp. 243-266 (ISSN: 0003-9527) | MR | Zbl | DOI

Le Coz, S.; Martel, Y.; Raphaël, P. Minimal mass blow up solutions for a double power nonlinear Schrödinger equation, Rev. Mat. Iberoam., Volume 32 (2016), pp. 795-833 (ISSN: 0213-2230) | DOI | MR

Martel, Y. Asymptotic $N$ -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., Volume 127 (2005), pp. 1103-1140 http://muse.jhu.edu/... (ISSN: 0002-9327) | MR | Zbl | DOI

Merle, F. Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., Volume 129 (1990), pp. 223-240 http://projecteuclid.org/euclid.cmp/1104180743 (ISSN: 0010-3616) | MR | Zbl | DOI

Merle, F. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., Volume 9 (1993), pp. 427-454 | MR | Zbl

Miot, E., Seminaire: Equations aux Dérivées Partielles. 2009–2010 (Sémin. Équ. Dériv. Partielles), École Polytech., Palaiseau, 2012 | MR | Zbl

Mizumachi, T. Weak interaction between solitary waves of the generalized KdV equations, SIAM J. Math. Anal., Volume 35 (2003), pp. 1042-1080 (ISSN: 0036-1410) | MR | Zbl | DOI

Martel, Y.; Merle, F. Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 23 (2006), pp. 849-864 (ISSN: 0294-1449) | MR | Zbl | mathdoc-id | DOI

Martel, Y.; Merle, F. Inelastic interaction of nearly equal solitons for the quartic gKdV equation, Invent. math., Volume 183 (2011), pp. 563-648 (ISSN: 0020-9910) | MR | Zbl | DOI

Martel, Y.; Merle, F.; Raphaël, P. Blow up for the critical generalized Korteweg de Vries equation. I: Dynamics near the soliton, Acta Math., Volume 212 (2014), pp. 59-140 (ISSN: 0001-5962) | MR | Zbl | DOI

Martel, Y.; Merle, F.; Raphaël, P. Blow up for the critical generalized Korteweg de Vries equation III: exotic regimes, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 14 (2015), pp. 575-631 (ISSN: 0391-173X) | MR

Martel, Y.; Merle, F.; Raphaël, P. Blow up for the critical generalized Korteweg de Vries equation. II: Minimal mass dynamics, J. of Math. Eur. Soc., Volume 17 (2015), pp. 1855-1925 | DOI

Martel, Y.; Merle, F.; Tsai, T.-P. Stability in $H^{1}$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations, Duke Math. J., Volume 133 (2006), pp. 405-466 | MR | Zbl | DOI

Musso, M.; Pacard, F.; Wei, J. Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., Volume 14 (2012), pp. 1923-1953 | MR | Zbl | DOI

Merle, F.; Raphaël, P. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal., Volume 13 (2003), pp. 591-642 (ISSN: 1016-443X) | MR | Zbl | DOI

Merle, F.; Raphaël, P. On universality of blow-up profile for $L^{2}$ critical nonlinear Schrödinger equation, Invent. math., Volume 156 (2004), pp. 565-672 (ISSN: 0020-9910) | MR | Zbl | DOI

Merle, F.; Raphaël, P. Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., Volume 253 (2005), pp. 675-704 (ISSN: 0010-3616) | MR | Zbl | DOI

Merle, F.; Raphaël, P. The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math., Volume 161 (2005), pp. 157-222 (ISSN: 0003-486X) | MR | Zbl | DOI

Merle, F.; Raphaël, P. On a sharp lower bound on the blow-up rate for the $L^{2}$ critical nonlinear Schrödinger equation, J. Amer. Math. Soc., Volume 19 (2006), pp. 37-90 (ISSN: 0894-0347) | MR | Zbl | DOI

Merle, F.; Raphaël, P.; Rodnianski, I. Type II blow up for the energy supercritical NLS, Camb. J. Math., Volume 3 (2015), pp. 439-617 (ISSN: 2168-0930) | DOI | MR

Merle, F.; Raphaël, P.; Szeftel, J. The instability of Bourgain-Wang solutions for the $L^{2}$ critical NLS, Amer. J. Math., Volume 135 (2013), pp. 967-1017 | 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.; Zaag, H. On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., Volume 333 (2015), pp. 1529-1562 | MR | Zbl | DOI

Nawa, H. Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math., Volume 52 (1999), pp. 193-270 (ISSN: 0010-3640) | MR | Zbl | 3.0.CO;2-3 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

Perelman, G. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, Volume 2 (2001), pp. 605-673 | MR | Zbl | DOI

Perelman, G. Two soliton collision for nonlinear Schrödinger equations in dimension 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 28 (2011), pp. 357-384 (ISSN: 0294-1449) | MR | Zbl | mathdoc-id | DOI

Planchon, F.; Raphaël, P. Existence and stability of the log-log blow-up dynamics for the $L^{2}$ -critical nonlinear Schrödinger equation in a domain, Ann. Henri Poincaré, Volume 8 (2007), pp. 1177-1219 (ISSN: 1424-0637) | MR | Zbl | DOI

Raphaël, P. Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., Volume 331 (2005), pp. 577-609 (ISSN: 0025-5831) | MR | Zbl | DOI

Raphaël, P.; Szeftel, J. Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., Volume 24 (2011), pp. 471-546 (ISSN: 0894-0347) | MR | Zbl | DOI

Weinstein, M. I. Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., Volume 87 (1982/83), pp. 567-576 http://projecteuclid.org/euclid.cmp/1103922134 (ISSN: 0010-3616) | MR | Zbl | DOI

Weinstein, M. I. Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., Volume 16 (1985), pp. 472-491 (ISSN: 0036-1410) | MR | Zbl | DOI

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