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Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. Part II

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Abstract

We consider the Zakharov equation in space dimension two

$$\left\{ {\begin{array}{*{20}c} {iu_t = - \Delta u + nu,} \\ {\frac{1}{{c_0^2 }}n_{tt} = \Delta n + \Delta \left| u \right|^2 } \\ \end{array} } \right.$$

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In the first part of the paper, we consider blow-up solutions of this equation. We prove various concentration properties of these solutions: existence, characterization of concentration mass, non existence of minimal concentration mass.

In the second part, we prove instability of periodic solutions.

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References

  1. Added, H., Added, S.: Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2. C.R. Acad. Sci. Paris299, 551–554 (1984)

    Google Scholar 

  2. Added, H., Added, S.: Equations of Langmuir turbulence and nonlinear Schrödinger equations: Smoothness and approximation. J. Funct. Anal.79, 183–210 (1988)

    Google Scholar 

  3. Berestycki, H., Cazenave, T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon nonlinéaires. C.R. Paris293, 489–492 (1981)

    Google Scholar 

  4. Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I. Existence of a ground state; II existence of infinitely many solutions. Arch. Rat. Mech. An.82, 313–375 (1983)

    Google Scholar 

  5. Brezis, H., Gallouet, T.: Non-linear Schrödinger evolution equations. Nonlinear Analysis, TMA4, 677–681 (1980)

    Google Scholar 

  6. Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations I, II. The Cauchy problem, general case. J. Funct. Anal.32, 1–71 (1979)

    Google Scholar 

  7. Glangetas, L., Merle, F.: Existence of self-similar blow-up solutions for Zakharov equation in dimension two, Part I. Commun. Math. Phys.160, 173–215 (1994)

    Google Scholar 

  8. Glassey, R.T.: On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation. J. Math. Phys.18, 1794–1797 (1977)

    Google Scholar 

  9. Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique49, 113–129 (1987)

    Google Scholar 

  10. Kwong, M.K.: Uniqueness of positive solution of Δu−u+u p=0 in ℝN. Arch. Rat. Mech. An.105, 243–266 (1989)

    Google Scholar 

  11. Landman, M., Papanicolaou, G.C., Sulem, C., Sulem, P.L., Wang, X.P.: Stability of isotropic self-similar dynamics for scalar collapse. Phys. Rev. A46, 7869–7876 (1992)

    Google Scholar 

  12. Lieb, E.H.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math.74, 441–448 (1983)

    Google Scholar 

  13. Lions, P.L.: The concentration-compactness principle in the calculus of variations: The locally compact case. Ann. Inst. Henri Poincaré, Analyse nonlinéaire1, 109–145 (1984)

    Google Scholar 

  14. Merle, F.: Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power. Duke J., to appear

  15. Merle, F.: On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Commun. Pure Appl. Math.45, 203–254 (1992)

    Google Scholar 

  16. Merle, F.: Sur la dépendance continue de la solution de l'équation de Schrödinger non lináire près du temps d'explosion. C.R. Paris16, 479–482 (1987)

    Google Scholar 

  17. Merle, F.: Limit behavior of saturated approximations of nonlinear Schrödinger equation. Commun. Math. Phys.149, 377–414 (1992)

    Google Scholar 

  18. Merle, F., Tsutsumi, Y.:L 2-concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity. J.D.E.84, 205–214 (1990)

    Google Scholar 

  19. Ozawa, T., Tsutsumi, Y.: The nonlinear Schrödinger Limit and the initial layer of the Zakharov Equations. Preprint

  20. Ozawa, T., Tsutsumi, Y.: Existence and smoothing effect of solutions for the Zakharov Equations. Preprint

  21. Papanicolaou, G.C., Sulem, C., Sulem, P.L., Wang, X.P.: Singular solutions of the Zakharov Equations for Langmuir turbulence. Phys. Fluids B3, 969–980 (1991)

    Google Scholar 

  22. Schochet, S.H., Weinstein, M.I.: The nonlinear limit of the Zakharov equations governing Langmuir turbulence. Commun. Math. Phys.106, 569–580 (1986)

    Google Scholar 

  23. Sobolev, V.V., Synakh, V.S., Zakharov, V.E.: Character of the singularity and stochastic phenomena in self-focusing. Zh. Eksp. Theor. Fiz, Pis'ma Red14, 390–393 (1971)

    Google Scholar 

  24. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149–162 (1977)

    Google Scholar 

  25. Sulem, C., Sulem, P.L.: Quelques résultats de régularité pour les équations de la turbulence de Langmuir. C.R. Acad. Paris289, 173–176 (1979)

    Google Scholar 

  26. Weinstein, M.I.: Modulational stability of ground states of the nonlinear Schrödinger equations. SIAM J. Math. Anal.16, 472–491 (1985)

    Google Scholar 

  27. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp Interpolation estimates. Commun. Math. Phys.87, 567–576 (1983)

    Google Scholar 

  28. Weinstein, M.I.: On the structure and formation of singularities in solutions to the nonlinear dispersive evolution equations. Commun. Partial Diff. Equ.11, 545–565 (1986)

    Google Scholar 

  29. Weinstein, M.I.: The nonlinear Schrödinger equation singularity formation stability and dispersion. AMS-SIAM Conference on the Connection between Infinite Dimensional and Finite Dimensional Dynamical Systems, July 1987

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Authors and Affiliations

  1. Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Tour 55-65, 5ème étage, 4, place Jussieu, F-75252, Paris Cedex 05, France

    L. Glangetas

  2. Centre de Mathématiques, Université de Cergy-Pontoise, Avenue du Parc, 8, Le Campus, F-95033, Cergy-Pontoise Cedex, France

    F. Merle

Authors
  1. L. Glangetas
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  2. F. Merle
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Additional information

Communicated by T. Spencer

This work was partially done while the second author was visiting Rutgers University and Courant Institute

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Glangetas, L., Merle, F. Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. Part II. Commun.Math. Phys. 160, 349–389 (1994). https://doi.org/10.1007/BF02103281

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  • DOI: https://doi.org/10.1007/BF02103281

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