Construction of blow-up solutions for Zakharov system on 𝕋 2
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, p. 791-824
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
Nous considérons le système de Zakharov dans lʼespace à deux dimensions avec la condition périodique au bord : {i t u=-Δu+nu, tt n=Δn+Δ|u| 2 ,(t,x)[0,T)×𝕋 2 .(Z) Nous prouvons lʼexistence de solutions de (Z) explosant au temps fini. En outre, nous prouvons quʼil nʼy a aucune solution explosive de masse minimale.
We consider the Zakharov system in two space dimension with periodic boundary condition: {i t u=-Δu+nu, tt n=Δn+Δ|u| 2 ,(t,x)[0,T)×𝕋 2 .(Z) We prove the existence of finite time blow-up solutions of (Z). Further, we show there exists no minimal mass blow-up solution.
@article{AIHPC_2013__30_5_791_0,
     author = {Kishimoto, Nobu and Maeda, Masaya},
     title = {Construction of blow-up solutions for Zakharov system on $ {\mathbb{T}}^{2}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {5},
     year = {2013},
     pages = {791-824},
     doi = {10.1016/j.anihpc.2012.09.003},
     zbl = {06295442},
     mrnumber = {3103171},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2013__30_5_791_0}
}
Kishimoto, Nobu; Maeda, Masaya. Construction of blow-up solutions for Zakharov system on $ {\mathbb{T}}^{2}$. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, pp. 791-824. doi : 10.1016/j.anihpc.2012.09.003. http://www.numdam.org/item/AIHPC_2013__30_5_791_0/

[1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Pure Appl. Math. (Amst.) vol. 140, Elsevier/Academic Press, Amsterdam (2003) | MR 2424078 | Zbl 0347.46040

[2] C. Antonini, Lower bounds for the L 2 minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Differential Integral Equations 15 (2002), 749-768 | MR 1893845 | Zbl 1016.35018

[3] I. Bejenaru, S. Herr, J. Holmer, D. Tataru, On the 2D Zakharov system with L 2 -Schrödinger data, Nonlinearity 22 (2009), 1063-1089 | MR 2501036 | Zbl 1173.35651

[4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107-156 | MR 1209299 | Zbl 0787.35097

[5] J. Bourgain, J. Colliander, On wellposedness of the Zakharov system, Int. Math. Res. Not. (1996), 515-546 | MR 1405972 | Zbl 0909.35125

[6] N. Burq, P. Gérard, N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal. 13 (2003), 1-19 | MR 1978490 | Zbl 1044.35084

[7] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10, New York University Courant Institute of Mathematical Sciences, New York (2003) | MR 2002047 | Zbl 1055.35003

[8] T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990), 807-836 | MR 1055532 | Zbl 0706.35127

[9] J. Ceccon, M. Montenegro, Optimal L p -Riemannian Gagliardo–Nirenberg inequalities, Math. Z. 258 (2008), 851-873 | MR 2369060 | Zbl 1139.46031

[10] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, arXiv:1104.1114v2 | MR 3406535 | Zbl 1331.35316

[11] J. Ginibre, Y. Tsutsumi, G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), 384-436 | MR 1491547 | Zbl 0894.35108

[12] L. Glangetas, F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I, Comm. Math. Phys. 160 (1994), 173-215 | MR 1262194 | Zbl 0808.35137

[13] L. Glangetas, 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 | MR 1262202 | Zbl 0808.35138

[14] R.T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797 | MR 460850 | Zbl 0372.35009

[15] N. Godet, Blow up in several points for the nonlinear Schrödinger equation on a bounded domain, Differential Integral Equations 24 (2011), 505-517 | MR 2809619 | Zbl 1249.35303

[16] J.L. Kelley, I. Namioka, Linear Topological Spaces, Grad. Texts in Math. vol. 36, Springer-Verlag, New York (1976) | MR 394084

[17] N. Kishimoto, Local well-posedness for the Zakharov system on multidimensional torus, J. Anal. Math., in press, arXiv:1109.3527v1. | MR 3043152

[18] S. Kwon, On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map, J. Differential Equations 245 (2008), 2627-2659 | MR 2455780 | Zbl 1153.35067

[19] F. Merle, P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), 591-642 | MR 1995801 | Zbl 1061.35135

[20] F. Merle, P. Raphael, On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), 565-672 | MR 2061329 | Zbl 1067.35110

[21] F. Merle, P. Raphael, On one blow up point solutions to the critical nonlinear Schrödinger equation, J. Hyperbolic Differ. Equ. 2 (2005), 919-962 | MR 2195987 | Zbl 1117.35075

[22] F. Merle, P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), 675-704 | MR 2116733 | Zbl 1062.35137

[23] F. Merle, P. Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2) 161 (2005), 157-222 | MR 2150386 | Zbl 1185.35263

[24] F. Merle, P. Raphael, On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), 37-90 | MR 2169042 | Zbl 1075.35077

[25] F. Merle, Y. Tsutsumi, L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations 84 (1990), 205-214 | MR 1047566 | Zbl 0722.35047

[26] H. Nawa, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math. 52 (1999), 193-270 | MR 1653454

[27] T. Ogawa, Y. Tsutsumi, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition, Functional-Analytic Methods for Partial Differential Equations, Tokyo, 1989, Lecture Notes in Math. vol. 1450, Springer, Berlin (1990), 236-251 | MR 1084613

[28] T. Ogawa, Y. Tsutsumi, Blow-up of H 1 solution for the nonlinear Schrödinger equation, J. Differential Equations 92 (1991), 317-330 | MR 1120908 | Zbl 0739.35093

[29] T. Ogawa, Y. Tsutsumi, Blow-up of H 1 solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc. 111 (1991), 487-496 | MR 1045145 | Zbl 0747.35004

[30] T. Ozawa, Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci. 28 (1992), 329-361 | MR 1184829 | Zbl 0842.35116

[31] P.A. Robinson, Nonlinear wave collapse and strong turbulence, Rev. Modern Phys. 69 (1997), 507-573

[32] J. Segata, Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrodinger type equation on torus, arXiv:1202.3211v1 | MR 2911420 | Zbl 1239.35150

[33] C. Sulem, P.-L. Sulem, The nonlinear Schrödinger equation, Self-Focusing and Wave Collapse, Appl. Math. Sci. vol. 139, Springer-Verlag, New York (1999) | MR 1696311 | Zbl 0659.35020

[34] H. Takaoka, N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on ×𝕋, J. Funct. Anal. 182 (2001), 427-442 | MR 1828800 | Zbl 0976.35085

[35] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/1983), 567-576 | MR 691044 | Zbl 0527.35023

[36] V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1972), 908-914 | MR 1131135