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Positive solutions for resonant (p, q)-equations with concave terms
Local well-posedness for the Zakharov system on the background of a line soliton
Fak. Mathematik, University of Vienna, Oskar MorgensternPlatz 1, A-1090 Wien, Austria |
We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove a weak convergence to a nonlinear Schrödinger equation.
References:
| [1] |
J. Ginibre, Y. Tsutsumi and G. Velo,
On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.
|
| [2] |
S. H. Schochet and M. I. Weinstein,
The nonlinear Schr¨odinger limit of the Zakharov equations governing Langmuir turbulence, Comm. math. Phys., 106 (1986), 569-580.
|
| [3] |
S. Klainerman and A. Majda,
Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524.
|
| [4] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53 (1984), Springer-Verlag, New York. |
| [5] |
D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188 (2013), American Mathematical Society, Providence, RI. |
| [6] |
C. Sulem and P. L. Sulem, The Nonlinear SchrÖdinger Equation, Applied Mathematical Sciences, 139 (1999), Springer-Verlag, New York. |
| [7] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
|
| [8] |
H. Takaoka,
Well-posedness for the Zakharov system with the periodic boundary condition, Differential and Integral Equations, 12 (1999), 789-810.
|
| [9] |
J. Bourgain,
On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.
|
| [10] |
L. Glangetas and F. Merle,
Existence of self-similar blow-up solutions for Zakharov equation in dimension two (Ⅰ), Comm. Math. Phys., 160 (1994), 173-215.
|
| [11] |
L. Glangetas and F. Merle,
Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two (Ⅱ), Comm. Math. Phys., 160 (1994), 349-389.
|
| [12] |
F. Rousset and N. Tzvetkov,
Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388.
|
| [13] |
F. Rousset and N. Tzvetkov,
Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496.
|
| [14] |
F. Rousset and N. Tzvetkov,
Transverse nonlinear instability of solitary waves for some Hamiltonian PDEs,, Math. Pures Appl. (9), 90 (2008), 550-590.
|
| [15] |
J. Bourgain and J. Colliander,
On wellposedness of the Zakharov system, Internat. math. Res. Notices, 11 (1996), 515-546.
|
| [16] |
T. Ozawa and Y. Tsutsumi,
Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361.
|
| [17] |
H. Added and S. Added,
Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 79 (1988), 183-210.
|
| [18] |
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
| [19] |
C. Sulem and P. Sulem,
Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris Sér. A-B, 289 (1979), A173-A176.
|
| [20] |
H. Added and S. Added,
Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554.
|
| [21] |
I. Bejenaru, S. Herr, J. Holmer and D. Tataru,
On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.
|
| [22] |
C. Kenig, G. Ponce and L. Vega,
On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal., 127 (1995), 204-234.
|
| [23] |
J. Simon,
Compact set in the Space Lp(0, T; B), Annali di Matematica pura ed applicata, (1987), 65-96.
|
show all references
References:
| [1] |
J. Ginibre, Y. Tsutsumi and G. Velo,
On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.
|
| [2] |
S. H. Schochet and M. I. Weinstein,
The nonlinear Schr¨odinger limit of the Zakharov equations governing Langmuir turbulence, Comm. math. Phys., 106 (1986), 569-580.
|
| [3] |
S. Klainerman and A. Majda,
Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524.
|
| [4] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53 (1984), Springer-Verlag, New York. |
| [5] |
D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188 (2013), American Mathematical Society, Providence, RI. |
| [6] |
C. Sulem and P. L. Sulem, The Nonlinear SchrÖdinger Equation, Applied Mathematical Sciences, 139 (1999), Springer-Verlag, New York. |
| [7] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.
|
| [8] |
H. Takaoka,
Well-posedness for the Zakharov system with the periodic boundary condition, Differential and Integral Equations, 12 (1999), 789-810.
|
| [9] |
J. Bourgain,
On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.
|
| [10] |
L. Glangetas and F. Merle,
Existence of self-similar blow-up solutions for Zakharov equation in dimension two (Ⅰ), Comm. Math. Phys., 160 (1994), 173-215.
|
| [11] |
L. Glangetas and F. Merle,
Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two (Ⅱ), Comm. Math. Phys., 160 (1994), 349-389.
|
| [12] |
F. Rousset and N. Tzvetkov,
Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388.
|
| [13] |
F. Rousset and N. Tzvetkov,
Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496.
|
| [14] |
F. Rousset and N. Tzvetkov,
Transverse nonlinear instability of solitary waves for some Hamiltonian PDEs,, Math. Pures Appl. (9), 90 (2008), 550-590.
|
| [15] |
J. Bourgain and J. Colliander,
On wellposedness of the Zakharov system, Internat. math. Res. Notices, 11 (1996), 515-546.
|
| [16] |
T. Ozawa and Y. Tsutsumi,
Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361.
|
| [17] |
H. Added and S. Added,
Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 79 (1988), 183-210.
|
| [18] |
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
| [19] |
C. Sulem and P. Sulem,
Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris Sér. A-B, 289 (1979), A173-A176.
|
| [20] |
H. Added and S. Added,
Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554.
|
| [21] |
I. Bejenaru, S. Herr, J. Holmer and D. Tataru,
On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.
|
| [22] |
C. Kenig, G. Ponce and L. Vega,
On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal., 127 (1995), 204-234.
|
| [23] |
J. Simon,
Compact set in the Space Lp(0, T; B), Annali di Matematica pura ed applicata, (1987), 65-96.
|
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