Asymptotic stability for self-similar blowup of mass-supercritical NLS

Zexing Li

doi:10.5802/slsedp.181

Zexing Li ¹

¹ Laboratoire Analyse, Géométrie et Modélisation, CY Cergy Paris Université, 2 avenue Adolphe Chauvin, 95300, Pontoise, France

Séminaire Laurent Schwartz — EDP et applications (2024-2025), Exposé no. 17, 13 p.

Résumé

We consider self-similar blowup for (NLS) $i \partial_{t} u + Δ u + u {| u |}^{p - 1} = 0$ in $d \geq 1$ , focusing on the slightly mass-supercritical range $0 < s_{c} : = \frac{d}{2} - \frac{2}{p - 1} ≪ 1$ . The existence and stability of such dynamics [39] and construction of suitable profiles [1] lead to the question of asymptotic stability. In this note, we review the background and recent results [25, 26, 27] on the asymptotic stability, with particular emphasis on mode stability and linear stability.

Publié le : 2025-09-23

DOI : 10.5802/slsedp.181

Affiliations des auteurs :

Zexing Li ¹

¹ Laboratoire Analyse, Géométrie et Modélisation, CY Cergy Paris Université, 2 avenue Adolphe Chauvin, 95300, Pontoise, France

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     title = {Asymptotic stability for self-similar blowup of mass-supercritical {NLS}},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
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     pages = {1--13},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2024-2025},
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Zexing Li. Asymptotic stability for self-similar blowup of mass-supercritical NLS. Séminaire Laurent Schwartz — EDP et applications (2024-2025), Exposé no. 17, 13 p.. doi: 10.5802/slsedp.181

Bibliographie
Cité par

[1] Yakine Bahri, Yvan Martel, and Pierre Raphaël. Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations. Ann. Henri Poincaré, 22(5):1701–1749, 2021. | DOI | Zbl

[2] Árpád Baricz. Turán type inequalities for modified Bessel functions. Bull. Aust. Math. Soc., 82(2):254–264, 2010. | DOI | Zbl

[3] Marius Beceanu. New estimates for a time-dependent Schrödinger equation. Duke Mathematical Journal, 159(3):417–477, 2011. | DOI | Zbl | MR

[4] V. S. Buslaev and G. S. Perelman. On the stability of solitary waves for nonlinear Schrödinger equations. In Nonlinear evolution equations, volume 164 of Amer. Math. Soc. Transl. Ser. 2, pages 75–98. American Mathematical Society, Providence, RI, 1995. | Zbl | DOI

[5] Rémi Carles. Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal., 35(4):823–843, 2003. | DOI | Zbl

[6] Thierry Cazenave. Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. | Zbl

[7] Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai. Spectra of linearized operators for NLS solitary waves. SIAM Journal on Mathematical Analysis, 39(4):1070–1111, 2008. | DOI | Zbl | MR

[8] Jiajie Chen and Thomas Y Hou. Stable nearly self-similar blowup of the 2d Boussinesq and 3d Euler equations with smooth data I: Analysis, 2022. | arXiv | Zbl

[9] Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi, and Van Tien Nguyen. Spectral analysis for singularity formation of the two dimensional Keller-Segel system. Ann. PDE, 8(1):Paper No. 5, 74, 2022. | Zbl | MR | DOI

[10] Charles Collot, Frank Merle, and Pierre Raphaël. Strongly anisotropic type II blow up at an isolated point. J. Amer. Math. Soc., 33(2):527–607, 2020. | DOI | Zbl | MR

[11] Charles Collot, Pierre Raphaël, and Jérémie Szeftel. On the stability of type I blow up for the energy super critical heat equation. Mem. Amer. Math. Soc., 260(1255):v+97, 2019. | Zbl

[12] Wei Dai and Thomas Duyckaerts. Self-similar solutions of focusing semi-linear wave equations in $ℝ^{N}$ . J. Evol. Equ., 21(4):4703–4750, 2021. | DOI | Zbl | MR

[13] Roland Donninger. Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation. Duke Math. J., 166(9):1627–1683, 2017. | DOI | Zbl | MR

[14] Roland Donninger and Birgit Schörkhuber. Self-similar blowup for the cubic Schrödinger equation, 2024. | arXiv | Zbl

[15] Tarek Elgindi. Finite-time singularity formation for $C^{1, α}$ solutions to the incompressible Euler equations on $ℝ^{3}$ . Ann. of Math. (2), 194(3):647–727, 2021. | DOI | Zbl

[16] Tarek M. Elgindi and Federico Pasqualotto. From instability to singularity formation in incompressible fluids, 2023. | arXiv | Zbl

[17] Gadi Fibich, Nir Gavish, and Xiao-Ping Wang. Singular ring solutions of critical and supercritical nonlinear Schrödinger equations. Phys. D, 231(1):55–86, 2007. | DOI | MR | Zbl

[18] Gadi Fibich, Frank Merle, and Pierre Raphaël. Proof of a spectral property related to the singularity formation for the $L^{2}$ critical nonlinear Schrödinger equation. Phys. D, 220(1):1–13, 2006. | DOI | Zbl | MR

[19] G. Guderley. Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung, 19:302–311, 1942. | Zbl | MR

[20] Justin Holmer, Galina Perelman, and Svetlana Roudenko. A solution to the focusing 3d NLS that blows up on a contracting sphere. Trans. Amer. Math. Soc., 367(6):3847–3872, 2015. | DOI | Zbl | MR

[21] Herbert Koch. Self-similar solutions to super-critical gKdV. Nonlinearity, 28(3):545–575, 2015. | DOI | Zbl

[22] J. Krieger and W. Schlag. Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Amer. Math. Soc., 19(4):815–920, 2006. | DOI | Zbl | MR

[23] M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang. Stability of isotropic singularities for the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 47(3):393–415, 1991. | DOI | Zbl

[24] B. J. LeMesurier, G. C. Papanicolaou, C. Sulem, and P. L. Sulem. Focusing and multi-focusing solutions of the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 31(1):78–102, 1988. | DOI | Zbl

[25] Zexing Li. On stability of self-similar blowup for mass supercritical NLS, 2023. | arXiv

[26] Zexing Li. Mode stability for self-similar blowup of slightly supercritical NLS: I. Low-energy spectrum, 2025 | arXiv

[27] Zexing Li. Mode stability for self-similar blowup of slightly supercritical NLS: II. High-energy spectrum, 2025. | arXiv

[28] Yvan Martel and Frank Merle. A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9), 79(4):339–425, 2000. | DOI | Zbl | MR

[29] Yvan Martel and Frank Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal., 157(3):219–254, 2001. | DOI | Zbl | MR

[30] Jeremy L. Marzuola and Gideon Simpson. Spectral analysis for matrix hamiltonian operators. Nonlinearity, 24(2):389, 2010. | DOI | Zbl

[31] David W. McLaughlin, George C. Papanicolaou, Catherine Sulem, and Pierre-Louis Sulem. Focusing singularity of the cubic Schrödinger equation. Physical Review A, 34(2):1200, 1986. | DOI

[32] Frank Merle and Pierre Raphaël. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal., 13(3):591–642, 2003. | DOI | Zbl

[33] Frank Merle and Pierre Raphaël. On universality of blow-up profile for $L^{2}$ critical nonlinear Schrödinger equation. Invent. Math., 156(3):565–672, 2004. | DOI | Zbl | MR

[34] Frank Merle and Pierre Raphaël. The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. (2), 161(1):157–222, 2005. | DOI | Zbl

[35] Frank Merle and Pierre Raphaël. Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Comm. Math. Phys., 253(3):675–704, 2005. | DOI | Zbl | MR

[36] Frank Merle and Pierre Raphaël. On a sharp lower bound on the blow-up rate for the $L^{2}$ critical nonlinear Schrödinger equation. J. Amer. Math. Soc., 19(1):37–90, 2006. | DOI | Zbl | MR

[37] Frank Merle and Pierre Raphaël. Blow up of the critical norm for some radial $L^{2}$ super critical nonlinear Schrödinger equations. Amer. J. Math., 130(4):945–978, 2008. | DOI | Zbl | MR

[38] Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jérémie Szeftel. On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles. Ann. of Math. (2), 196(2):567–778, 2022. | DOI | Zbl | MR

[39] Frank Merle, Pierre Raphaël, and Jérémie Szeftel. Stable self-similar blow-up dynamics for slightly $L^{2}$ super-critical NLS equations. Geom. Funct. Anal., 20(4):1028–1071, 2010. | DOI | Zbl

[40] Frank Merle, Pierre Raphaël, and Jérémie Szeftel. On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation. Duke Math. J., 163(2):369–431, 2014. | DOI | Zbl

[41] Galina Perelman. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré, 2(4):605–673, 2001. | DOI | Zbl

[42] Pierre Raphaël. Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation. Math. Ann., 331(3):577–609, 2005. | DOI | Zbl

[43] Pierre Raphaël. Existence and stability of a solution blowing up on a sphere for an $L^{2}$ -supercritical nonlinear Schrödinger equation. Duke Math. J., 134(2):199–258, 2006. | DOI | Zbl

[44] Pierre Raphaël. On the singularity formation for the nonlinear Schrödinger equation. Clay Math. Proc, 17:269–323, 2013. | Zbl

[45] Michael Reed and Barry Simon. Methods of modern mathematical physics. III. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. | Zbl

[46] W. Schlag. Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. Ann. of Math. (2), 169(1):139–227, 2009. | DOI | Zbl

[47] Catherine Sulem and Pierre-Louis Sulem. Focusing nonlinear Schrödinger equation and wave-packet collapse. In Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996), volume 30, pages 833–844, 1997. | DOI | Zbl | MR

[48] William C Troy. Zero energy self-similar solutions describing singularity formation in the nonlinear Schrödinger equation in dimension $n = 3$ , 2022. | arXiv | Zbl

[49] Kai Yang, Svetlana Roudenko, and Yanxiang Zhao. Blow-up dynamics and spectral property in the $L^{2}$ -critical nonlinear Schrödinger equation in high dimensions. Nonlinearity, 31(9):4354–4392, 2018. | DOI | MR | Zbl

[50] Kai Yang, Svetlana Roudenko, and Yanxiang Zhao. Blow-up dynamics in the mass super-critical NLS equations. Phys. D, 396:47–69, 2019. | DOI | Zbl | MR

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