ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA*

doi:10.1007/s10473-023-0517-5

Abstract

Abstract: We consider the fourth-order nonlinear Schrödinger equation (4NLS) \begin{align*} ({\rm i}\partial_t+\varepsilon\Delta+\Delta^2)u=c_1u^m+c_2(\partial u)u^{m-1}+c_3(\partial u)^2u^{m-2}, \end{align*} and establish the conditional almost sure global well-posedness for random initial data in $H^s(\mathbb{R}^d)$ for $s\in (s_c-1/2, \ s_c]$, when $d\geq3$ and $m\geq5$, where $s_c:=d/2-2/(m-1)$ is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new $M$-norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for $d=2, \ m\geq4$ and $ d\geq3, \ 3\leq m<5$.

Key words: fourth-order Schrödinger equation, random initial data, almost sure global well-posedness, $M$-norm, stability theory

CLC Number:

35Q55

Mingjuan Chen, Shuai Zhang. ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA^*[J].Acta mathematica scientia,Series A, 2023, 43(5): 2215-2233.

Trendmd

References

[1] Ben-Artzi M, Koch H, Saut J C. Dispersion estimates for fourth order Schrödinger equations. C R Acad Sci Paris Sér I Math, 2000, 330(2): 87-92
[2] Bényi Á, Oh T, Pocovnicu O. Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS//Balan R, Begue M, Benedetto J, et al. Excursions in Harmonic Analysis, Volume 4. Applied and Numerical Harmonic Analysis. Switzerland: Birkhäuser, 2015: 3-25
[3] Bényi Á, Oh T, Pocovnicu O. On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^d, \ d\geq3 $. Trans Amer Math Soc Ser B, 2015, 2(1): 1-50
[4] Bourgain J. Periodic nonlinear Schrödinger equation and invariant measures. Comm Math Phys, 1994, 166(1): 1-26
[5] Bourgain J. Invariant measures for the {$2$}D-defocusing nonlinear Schrödinger equation. Comm Math Phys, 1996, 176(2): 421-445
[6] Bringmann B. Almost sure scattering for the energy critical nonlinear wave equation. Amer J Math, 2021, 143(6): 1931-1982
[7] Burq N, Tzvetkov N. Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent Math, 2008, 173(3): 449-475
[8] Burq N, Tzvetkov N. Random data Cauchy theory for supercritical wave equations. II. A global existence result. Invent Math, 2008, 173(3): 477-496
[9] Chen M, Zhang S. Random data cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities. Nonlinear Anal, 2020, 190: Art 111608
[10] Christ F M, Weinstein M I. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J Funct Anal, 2013, 16(1): 1-30
[11] Da Prato G, Debussche A. Two-dimensional Navier-Stokes equations driven by a space-time white noise. J Funct Anal, 2002, 196: 180-210
[12] Deng Y, Nahmod A, Yue H.Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two. arXiv: 1910.08492
[13] Deng Y, Nahmod A, Yue H. Random tensors, propagation of randomness, nonlinear dispersive equations. Invent Math, 2022, 228(2): 539-686
[14] Dinh V D. Well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation. Bull Belg Math Soc Simon Stevin, 2018, 25: 415-437
[15] Dodson B Lührmann J, Mendelson D. Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data. Amer J Math, 2020, 142(2): 475-504
[16] Dodson B Lührmann J, Mendelson D. Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation. Adv Math, 2019, 347: 619-676
[17] Dysthe K B. Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc R Soc Lond A, 1979, 369(1736): 105-114
[18] Hadac M, Herr S, Koch H. Well-posedness and scattering for the KP-II equation in a critical space. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26(3): 917-941
[19] Herr S, Tataru D, Tzvetkov N. Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in {$H^1(\mathbb{T}^3)$}. Duke Math J, 2011, 159(2): 329-349
[20] Hirayama H, Okamoto M. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete Contin Dyn Syst, 2016, 36(12): 6943-6974
[21] Huo Z H, Jia Y L. A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament. Comm Partial Differential Equations, 2007, 32(10): 1493-1510
[22] Ionescu A D, Pausader B. Global well-posedness of the energy-critical defocusing NLS on {$\mathbb{R}\times \mathbb{T}^3$}. Comm Math Phys, 2012, 312(3): 781-831
[23] Ilan B, Fibich G, Papanicolaou G. Self-focusing with fourth-order dispersion. SIAM J Appl Math, 2002, 62(4): 1437-1462
[24] Karpman V I. Stabilization of soliton instabilities by higher-order dispersion: fourth-order nonlinear Schrödinger-type equations. Physical Review E, 1996, 53(2): 1336-1339
[25] Karpman V I, Shagalov A G. Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Phys D, 2000, 144(1/2): 194-210
[26] Kenig C, Mendelson D.The focusing energy-critical nonlinear wave equation with random initial data. Int Math Res Not, 2021, (19): 14508-14615
[27] Killip R, Murphy J, Visan M. Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb{R}^4)$. Comm Partial Differential Equations, 2019, 44(1): 51-71
[28] Lührmann J, Mendelson D. Random data Cauchy theory for nonlinear wave equations of power-type on {$\mathbb{R}^3$}. Comm Partial Differential Equations, 2014, 39(12): 2262-2283
[29] Mendelson D. Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on {$\mathbb{T}^3$}. J Funct Anal, 2017, 272(7): 3019-3092
[30] Pausader B. Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn Partial Differ Equ, 2007, 4(3): 197-225
[31] Pausader B. The cubic fourth-order Schrödinger equation. J Funct Anal, 2009, 256(8): 2473-2517
[32] Pocovnicu O. Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$. J Eur Math Soc, 2017, 19(8): 2521-2575
[33] Ren Y Y, Li Y S, Wei Y. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Commun Pure Appl Anal, 2018, 17(2): 487-504
[34] Ruzhansky M, Wang B X, Zhang H. Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces. J Math. Pures Appl, 2016, 105(1): 31-65
[35] Stein E M.Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993
[36] Tao T.Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; Providence, RI: American Mathematical Society, 2006
[37] Zhang S, Xu S P. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Commun Pure Appl Anal, 2020, 19(6): 3367-3385

ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA^*

Abstract

Cite this article

share this article

References

Related Articles 11

Metrics

Comments

Recommended 0

[1]	LIANG Yun-Yun, LI Chu-Jin, HAO Cai-Di. Compact Kernel Sections and Kolmogorov Entropy for the Lattice Zakharov Equations with a Quantum Correction [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1203-1218.
[2]	HUANG Juan, ZHANG Jian. The Weakly Damped Nonlinear Hartree Equation [J]. Acta mathematica scientia,Series A, 2014, 34(2): 259-265.
[3]	YANG Xin-Bo, ZHAO Cai-De, JIA Xiao-Lin. The Uniform Attractor and Entropy of the Autonomous Coupled Nonlinear SchrÖdinger Equations on Infinite Lattices [J]. Acta mathematica scientia,Series A, 2013, 33(4): 636-645.
[4]	CHEN Shu-Hong, GUO Bo-Ling. The Existence of Global Solution of the Ginzburg-Landau Equations for Atomic Fermi Gases Near the BCS-BEC Crossover [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1359-1368.
[5]	ZHAO Cai-De. H¹-uniform Attractor for 2D Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1416-1430.
[6]	ZHANG Xiao-Yi. The Self similar Solution and its Limit Behavior for the Complex Ginzburg Landau Equation [J]. Acta mathematica scientia,Series A, 2005, 25(5): 652-662.
[7]	SHANG Y-Dong, GUO Bo-Ling. Approximate Inertial Manifolds for the Nonlinear Sobolev Galpern Equations [J]. Acta mathematica scientia,Series A, 2004, 24(1): 105-115.
[8]	FANG Shao-Mei, GUO BO-Ling. Attractors for a Damped Generalized Coupled Nonlinear Wave Equations in an Unbounded Domain [J]. Acta mathematica scientia,Series A, 2003, 23(4): 464-473.
[9]	CHEN Han-Lin, DAI Zheng-De. Exponential Attrctor of the Cauchy Problem of Disspative Zakharov Equation in R [J]. Acta mathematica scientia,Series A, 2001, 21(3): 373-383.
[10]	LIU Xiao-Juan, HAI Wen-Hua, FANG Jian-Shu, ZHANG Xi-Li. Using Classical Chaos To Simulate Quantum Uncomputable State [J]. Acta mathematica scientia,Series A, 2001, 21(1): 43-47.
[11]	Du Xianyun Dai Zhengde. Attractor for Cauchy Problem of Dissiptive KDV type Equation [J]. Acta mathematica scientia,Series A, 2000, 20(3): 289-295.