May  2019, 39(5): 2661-2678. doi: 10.3934/dcds.2019111

Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential

1. 

Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland

2. 

Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri, 3, 56126 Pisa, Italy

3. 

Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 I - 56127 Pisa, Italy

4. 

Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

5. 

IMI-BAS, Acad, Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria

6. 

Department of Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

* Corresponding author: Luigi Forcella

Received  May 2018 Revised  November 2018 Published  January 2019

We study the initial value problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.

Citation: Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
References:
[1]

P. AcquistapaceA. P. CandeloroV. Georgiev and M. L. Manca, Mathematical phase model of neural populations interaction in modulation of REM/NREM sleep, Math. Model. Anal., 21 (2016), 794-810.  doi: 10.3846/13926292.2016.1247302.

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.

[3]

M. BravermanO. Milatovich and M. Shubin, Essential selfadjointness of Schródinger-type operators on manifolds, Uspekhi Mat. Nauk, 57 (2002), 3-58.  doi: 10.1070/RM2002v057n04ABEH000532.

[4] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. 
[5]

K. Fujiwara, V. Georgiev and T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases, arXiv: 1611.09674.

[6]

K. FujiwaraV. Georgiev and T. Ozawa, Blow-up for self-interacting fractional Ginzburg-Landau equation, Dyn. Partial Differ. Equ., 15 (2018), 175-182.  doi: 10.4310/DPDE.2018.v15.n3.a1.

[7]

V. GeorgievA. Michelangeli and R. Scandone, On fractional powers of singular perturbations of the Laplacian, J. Funct. Anal., 275 (2018), 1551-1602.  doi: 10.1016/j.jfa.2018.03.007.

[8]

L. Grafakos and S. Oh, The Kato-Ponce Inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.

[9]

E. Heinz, Beiträge zur Stórungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438.  doi: 10.1007/BF02054965.

[10]

R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997, Elementary theory, Reprint of the 1983 original.

[11]

T. Kappeler, P. Perry, M. Shubin and P. Topalov, The Miura map on the line, Int. Math. Res. Not., (2005), 3091–3133. doi: 10.1155/IMRN.2005.3091.

[12]

T. Kato, Notes on some inequalities for linear operators, Math. Ann., 125 (1952), 208-212.  doi: 10.1007/BF01343117.

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[14]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[15]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[16]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[17]

D. Li, On Kato-Ponce and fractional Leibniz, arXiv: 1609.01780, (2016), to appear in Rev. Mat. Iberoamericana.

[18]

K. Löwner, Über monotone Matrixfunktionen. (German), Trans. Amer. Math. Soc., 38 (1934), 177-216.  doi: 10.1007/BF01170633.

[19]

G. K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc., 36 (1972), 309-310.  doi: 10.2307/2039083.

[20] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅰ. Functional Analysis, Academic Press, New York-London, 1972. 
[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[22]

V. Tarasov and G. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2016), 023110, 13pp. doi: 10.1063/1.2197167.

[23]

M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.

[24]

W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D, 56 (1992), 303-367.  doi: 10.1016/0167-2789(92)90175-M.

show all references

References:
[1]

P. AcquistapaceA. P. CandeloroV. Georgiev and M. L. Manca, Mathematical phase model of neural populations interaction in modulation of REM/NREM sleep, Math. Model. Anal., 21 (2016), 794-810.  doi: 10.3846/13926292.2016.1247302.

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.

[3]

M. BravermanO. Milatovich and M. Shubin, Essential selfadjointness of Schródinger-type operators on manifolds, Uspekhi Mat. Nauk, 57 (2002), 3-58.  doi: 10.1070/RM2002v057n04ABEH000532.

[4] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. 
[5]

K. Fujiwara, V. Georgiev and T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases, arXiv: 1611.09674.

[6]

K. FujiwaraV. Georgiev and T. Ozawa, Blow-up for self-interacting fractional Ginzburg-Landau equation, Dyn. Partial Differ. Equ., 15 (2018), 175-182.  doi: 10.4310/DPDE.2018.v15.n3.a1.

[7]

V. GeorgievA. Michelangeli and R. Scandone, On fractional powers of singular perturbations of the Laplacian, J. Funct. Anal., 275 (2018), 1551-1602.  doi: 10.1016/j.jfa.2018.03.007.

[8]

L. Grafakos and S. Oh, The Kato-Ponce Inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.

[9]

E. Heinz, Beiträge zur Stórungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438.  doi: 10.1007/BF02054965.

[10]

R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997, Elementary theory, Reprint of the 1983 original.

[11]

T. Kappeler, P. Perry, M. Shubin and P. Topalov, The Miura map on the line, Int. Math. Res. Not., (2005), 3091–3133. doi: 10.1155/IMRN.2005.3091.

[12]

T. Kato, Notes on some inequalities for linear operators, Math. Ann., 125 (1952), 208-212.  doi: 10.1007/BF01343117.

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[14]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[15]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[16]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[17]

D. Li, On Kato-Ponce and fractional Leibniz, arXiv: 1609.01780, (2016), to appear in Rev. Mat. Iberoamericana.

[18]

K. Löwner, Über monotone Matrixfunktionen. (German), Trans. Amer. Math. Soc., 38 (1934), 177-216.  doi: 10.1007/BF01170633.

[19]

G. K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc., 36 (1972), 309-310.  doi: 10.2307/2039083.

[20] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅰ. Functional Analysis, Academic Press, New York-London, 1972. 
[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[22]

V. Tarasov and G. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2016), 023110, 13pp. doi: 10.1063/1.2197167.

[23]

M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.

[24]

W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D, 56 (1992), 303-367.  doi: 10.1016/0167-2789(92)90175-M.

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