Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. (English) Zbl 0808.35137

Glangetas, L.; Merle, F.

Commun. Math. Phys. 160, No. 1, 173-215 (1994).

See the review of part II below.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B40	Asymptotic behavior of solutions to PDEs
76X05	Ionized gas flow in electromagnetic fields; plasmic flow

Keywords:

blow-up solutions; concentration properties; existence; concentration mass; instability of periodic solutions

Citations:

Zbl 0808.35138

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. Paris299, 551–554 (1984) · Zbl 0575.35080
[2]	Added, H., Added, S.: Equations of Langmuir turbulence and nonlinear Schrödinger equations. Smoothness and approximation. J. Funct. Anal.79, 183–210 (1988) · Zbl 0655.76044 · doi:10.1016/0022-1236(88)90036-5
[3]	Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. I. Exsistence of a ground state; II existence of infinitely many solutions. Arch. Rat. Mech. An.82, 313–375 (1983) · Zbl 0556.35046
[4]	Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys.85, 549–561 (1982) · Zbl 0513.35007 · doi:10.1007/BF01403504
[5]	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) · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90076-4
[6]	Glangetas, L., Merle, F.: Concentration properties of blow-up solutions and instability results for Zakharov equations in dimension two, Part II. Commun. Math. Phys., to appear · Zbl 0808.35138
[7]	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) · Zbl 0372.35009 · doi:10.1063/1.523491
[8]	Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique49, 113–129 (1987)
[9]	Kwong, M.K.: Uniqueness of positive solutions of {\(\Delta\)}u+u p =0 in \(\mathbb{R}\)N, Arch. Rat. Mech. An.105, 243–266 (1989) · Zbl 0676.35032
[10]	Landman, M., Papanicolaou, G.C., Sulen, C., Sulen, P.L., Wang, X.P.: Stability of isotropic self-similar dynamics for scalar collapse. Phys. Rev. A46, 7869–7876 (1992) · doi:10.1103/PhysRevA.46.7869
[11]	Merle, F.: Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power. Duke J. (to appear) · Zbl 0808.35141
[12]	Merle, F.: On uniqueness and continutation 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) · Zbl 0767.35084 · doi:10.1002/cpa.3160450204
[13]	Ozawa, T., Tsutsumi, Y.: The nonlinear Schrödinger Limit and the initial layer of the Zakharov equations. Preprint · Zbl 0754.35113
[14]	Ozawa, T., Tsutsumi, Y.: Existence and smoothing effect of solutions pfor the Zakharov equations. Preprint · Zbl 0842.35116
[15]	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) · doi:10.1063/1.859852
[16]	Pohozaev, S.I.: Eigenfunctions of the equation {\(\Delta\)}u+f(u)=0. Soviet Math.5 1408–1411 (1965) · Zbl 0141.30202
[17]	Rabinowitz, P.H.: Some aspect of nonlinear eigenvalue problems. Rocky Mountain J. Math.3, 161–202 (1973) · Zbl 0255.47069 · doi:10.1216/RMJ-1973-3-2-161
[18]	Schochet, S.H., Weinstein, D.H.I.: The nonlinear limit of the Zakharov equations governing Langmuir turbulence. Commun. Math. Phys.106, 569–580 (1986) · Zbl 0639.76054 · doi:10.1007/BF01463396
[19]	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)
[20]	Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149–162 (1977) · Zbl 0356.35028 · doi:10.1007/BF01626517
[21]	Sulem, C., Sulem, P.L.: Quelques résultats de r’egularité pour les équations de la turbulence de Langmuir. C.R. Acad. Sci. Paris289, 173–176 (1979) · Zbl 0431.35077
[22]	Weinstein, M.I.: Modulation stability of ground states of the nonlinear Schrödinger equations. SIAM J.Math. Anal.16, 472–491 (1985) · Zbl 0583.35028 · doi:10.1137/0516034
[23]	Weinstein, M.I.: Nonlinear Schrödinger equations and sharp Interpolation estimates. Commun. Math. Phys.87, 567–576 (1983) · Zbl 0527.35023 · doi:10.1007/BF01208265