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On Effective PDEs of Quantum Physics Read chapter Read first chapter

2019 | OriginalPaper | Chapter

Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case

Authors : Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa

Published in: New Tools for Nonlinear PDEs and Application

Publisher: Springer International Publishing

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Abstract

The aim of this work is to give a complete picture concerning the asymptotic behaviour of the solutions to fractional Ginzburg-Landau equation. In previous works, we have shown global well-posedness for the past interval in the case where spatial dimension is less than or equal to 3. Moreover, we have also shown blow-up of solutions for the future interval in one dimensional case. In this work, we summarise the asymptotic behaviour in the case where spatial dimension is less than or equal to 3 by proving blow-up of solutions for a future time interval in multidimensional case. The result is obtained via ODE argument by exploiting a new weighted commutator estimate.

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previous chapter The Critical Exponent for Evolution Models with Power Non-linearity
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Literature
1.
J. Bellazzini, V. Georgiev, N. Visciglia, Long time dynamics for semirelativistic NLS and half wave in arbitrary dimension. Math. Annalen. 371(1–2), 707–740 (2018)MathSciNetCrossRef
2.
J.P. Borgna, D.F. Rial, Existence of ground states for a one-dimensional relativistic Schödinger equation. J. Math. Phys. 53, 062301 (2012)MathSciNetCrossRef
3.
4.
T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10 (American Mathematical Society/New York University/Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003)
5.
T. Cazenave, F.B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations and Attractors, ed. by T.L. Gill, W.W. Zachary (Washington, DC, 1987). Lecture Notes in Mathematics, vol. 1394 (Springer, Berlin, 1989), pp. 18–29
6.
T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear analysis. Theory, methods & applications. Int. Multidiscip. J. Ser. A Theory Methods 14, 807–836 (1990)
7.
8.
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRef
9.
L. Forcella, K. Fujiwara, V. Georgiev, T. Ozawa, Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete Contin. Dyn. Syst. 39, 2661–2678 (2019). arXiv:1804.02524
10.
K. Fujiwara, Remark on local solvability of the Cauchy problem for semirelativistic equations. J. Math. Anal. Appl. 432, 744–748 (2015)MathSciNetCrossRef
11.
K. Fujiwara, V. Georgiev, T. Ozawa, Blow-up for self-interacting fractional Ginzburg-Landau equation. Dyn. Partial Differ. Equ. 15, 175–182 (2018)MathSciNetCrossRef
12.
K. Fujiwara, V. Georgiev, T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases. arXiv: 1611.09674 (2016)
13.
J. Ginibre, T. Ozawa, G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60, 211–239 (1994)MathSciNetMATH
14.
J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)MathSciNetCrossRef
15.
16.
M. Ikeda, T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance. J. Math. Anal. Appl. 425, 758–773 (2015)MathSciNetCrossRef
17.
M. Ikeda, Y. Wakasugi, Small-data blow-up of L 2-solution for the nonlinear Schrödinger equation without gauge invariance. Differ. Integral Equ. 26, 11–12 (2013)MATH
18.
T. Inui, Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity. Proc. Am. Math. Soc. 144, 2901–2909 (2016)CrossRef
19.
T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)MathSciNetCrossRef
20.
C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993)MathSciNetCrossRef
21.
S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993)MathSciNetCrossRef
22.
A. Kufner, B. Opic, Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series (Longman Scientific & Technical, Harlow, 1990)
23.
24.
E. Lenzmann, A. Schikorra, Sharp commutator estimates via harmonic extensions. arXiv: 1609.08547
25.
D. Li, On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoamericana. (in press). arXiv:1609.01780v2. It appeared on arXiv in 2016 and revised in 2018
26.
M. Nakamura, T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces. Publ. Res. Inst. Math. Sci. 37, 255–293 (2001)MathSciNetCrossRef
27.
T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 25, 403–408 (2006)CrossRef
28.
W. Sickel, L. Skrzypczak, Radial subspaces of Besov and Lizorkin-Triebel classes: extended Strauss lemma and compactness of embeddings. J. Fourier Anal. Appl. 6, 639–662 (2000)MathSciNetCrossRef
Metadata
Title
Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case
Authors
Luigi Forcella
Kazumasa Fujiwara
Vladimir Georgiev
Tohru Ozawa
Copyright Year
2019
Book
New Tools for Nonlinear PDEs and Application

Print ISBN: 978-3-030-10936-3

Electronic ISBN: 978-3-030-10937-0

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-10937-0_6

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