On the long time behavior of KdV type equations
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 933, pp. 219-248.

In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg-de-Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.

Dans une série d'articles récents, Martel et Merle ont mis en évidence l'existence de solutions qui explosent en temps fini, dans l'espace d'énergie, pour l'équation de KdV généralisée critique, résolvant ainsi une conjecture ancienne. Ils ont introduit des outils nouveaux pour étudier la dynamique non linéaire au voisinage d'une onde solitaire. Le but de cet exposé est de présenter les idées principales développées par Martel-Merle.

Classification: 35Q53,  35B35
Keywords: explosion en temps fini, EDP hamiltonienne, KdV
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Tzvetkov, Nikolay. On the long time behavior of KdV type equations, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 933, pp. 219-248. http://archive.numdam.org/item/SB_2003-2004__46__219_0/

[1] M. J. Ablowitz and H. Segur. Solitons and the inverse scattering transform. SIAM, Philadelphia, 1981. | MR | Zbl

[2] S. Alinhac. Blow-up for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Boston, 1995. | MR | Zbl

[3] T. Benjamin. Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 29:559-592, 1967. | Zbl

[4] T. Benjamin. The stability of solitary waves. Proc. London Math. Soc. (3), 328:153-183, 1972. | MR

[5] T. Benjamin, J. Bona, and J. Mahony. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A, 272:47-78, 1972. | MR | Zbl

[6] J. Bona. The stability of solitary waves. Proc. London Math. Soc. (3), 344:363-374, 1975. | MR | Zbl

[7] J. Bona and R. Smith. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A, 278:555-601, 1975. | MR | Zbl

[8] J. Bona, P. Souganidis, and W. Strauss. Stability and instability of solitary waves of Korteweg-de Vries type. Proc. London Math. Soc. (3), 411:395-412, 1987. | MR | Zbl

[9] J. Bona and F. Weissler. Similarity solutions of the generalized Korteweg-de Vries equation. Math. Proc. Cambridge Philos. Soc., 127:323-351, 1999. | MR | Zbl

[10] A. De Bouard and Y. Martel. Non existence of ${L}^{2}$-compact solutions of the Kadomtsev-Petviashvili II equation. Math. Ann., 328:525-544, 2004. | MR

[11] J. Bourgain. Global solutions of nonlinear Schrödinger equations, volume 46 of AMS Colloquium Publications. American Mathematical Society, Providence, R.I., 1999. | MR | Zbl

[12] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV equation. Geom. Funct. Anal., 3:209-262, 1993. | MR | Zbl

[13] T. Cazenave. Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes. 2003. | MR | Zbl

[14] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrödinger equation. Comm. Math. Phys., 85:549-561, 1982. | MR | Zbl

[15] J.-Y. Chemin. Explosion géométrique pour certaines équations d'ondes non linéaires (d'après Serge Alinhac). In Sém. Bourbaki (1998/99), volume 266 of Astérisque, pages 7-20. Société Mathématique de France, 2000. Exp. 850. | Numdam | MR | Zbl

[16] W. Eckhaus and P. Schuur. The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions. Math. Methods Appl. Sci., 5:97-116, 1983. | MR | Zbl

[17] K. El Dika. Stabilité asymptotique des ondes solitaires de l'équation de Benjamin-Bona-Mahony. C. R. Acad. Sci. Paris Sér. I Math., 337:649-652, 2003. | MR | Zbl

[18] K. El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Preprint, 2003. | MR | Zbl

[19] C. Fermanian, F. Merle, and H. Zaag. Stability of the blow-up profile of non-linear heat equations from a dynamical system point of view. Math. Ann., 317:347-387, 2000. | MR | Zbl

[20] S. Friedlander, W. Strauss, and M. Vishik. Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 14:187-209, 1997. | Numdam | MR | Zbl

[21] J. Ginibre and Y. Tsutsumi. Uniqueness of solutions for the generalized Korteweg-de Vries equation. SIAM J. Appl. Math., 20:1388-1425, 1989. | MR | Zbl

[22] L. Glangetas and F. Merle. A geometric approach of existence of blow-up solutions. Preprint, 1995.

[23] M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal., 74:160-197, 1987. | MR | Zbl

[24] L. Hörmander. The analysis of linear partial differential operators I. Springer-Verlag, 1983. | Zbl

[25] T. Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. volume 8 of Advances in Math. Suppl. Stud., pages 93-128. 1983. | MR | Zbl

[26] C. Kenig and K. Koenig. On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett., 10:879-895, 2003. | MR | Zbl

[27] C. Kenig, G. Ponce, and L. Vega. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46:527-620, 1993. | MR | Zbl

[28] C. Kenig, G. Ponce, and L. Vega. On the concentration of blow-up solutions for the generalized KdV equation critical in ${L}^{2}$. volume 263 of Contemp. Math., pages 131-156. American Mathematical Society, 2000. | MR | Zbl

[29] G. L. Lamb Jr. Elements of soliton theory. John Wiley & Sons, New York, 1980. | MR | Zbl

[30] C. Laurent and Y. Martel. Smoothness and exponential decay of ${L}^{2}$-compact solutions of the generalized KdV equation. Comm. Partial Differential Equations, 28:2093-2107, 2003. | MR | Zbl

[31] J. Maddocks and R. Sachs. On the stability of KdV multi-solitons. Comm. Pure Appl. Math., 46:867-901, 1993. | MR | Zbl

[32] Y. Martel. Multi-soliton-type solutions of the generalized KdV equations. Amer. J. Math. to appear. | MR | Zbl

[33] Y. Martel and F. Merle. Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal., 11:74-123, 2001. | MR | Zbl

[34] Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rational Mech. Anal., 157:219-254, 2001. | MR | Zbl

[35] Y. Martel and F. Merle. A Liouville Theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl., 79:339-425, 2000. | MR | Zbl

[36] Y. Martel and F. Merle. Stability of the blow-up profile and lower bounds on the blow-up rate for the critical generalized Korteweg-de Vries equation. Ann. of Math., 155:235-280, 2002. | MR | Zbl

[37] Y. Martel and F. Merle. Blow-up in finite time and dynamics of blow-up solutions for the ${L}^{2}$-critical generalized KdV equation. J. Amer. Math. Soc., 15:617-663, 2002. | MR | Zbl

[38] Y. Martel and F. Merle. Nonexistence of blow-up solution with minimal ${L}^{2}$-mass for the critical GKdV. Duke Math. J., 115:385-408, 2002. | MR | Zbl

[39] Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations revisited. Preprint, 2004. | MR | Zbl

[40] Y. Martel, F. Merle, and Tai-Peng Tsai. Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations. Comm. Math. Phys., 231:347-373, 2002. | MR | Zbl

[41] F. Merle. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J., 69:427-454, 1993. | MR | Zbl

[42] F. Merle. Asymptotics for ${L}^{2}$-minimal blow-up solutions of critical nonlinear Schrödinger equation. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13:553-565, 1996. | Numdam | MR | Zbl

[43] F. Merle. Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations. In Proceeding of the International congress of Mathematicians (Berlin 1998), Doc. Math. Extra volume ICM (1998 III), pages 57-66. Deutsche Math. Vereinigung, 1998. | MR | Zbl

[44] F. Merle. Existence of blow-up solutions in the energy space for critical generalized KdV equation. J. Amer. Math. Soc., 14:555-578, 2001. | MR | Zbl

[45] F. Merle and P. Raphaël. Sharp upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Geom. Funct. Anal., 13:591-642, 2003. | MR | Zbl

[46] F. Merle and P. Raphaël. On universality of blow-up profile for ${L}^{2}$-critical nonlinear Schrödinger equation. Invent. Math., 156:565-672, 2004. | MR | Zbl

[47] F. Merle and P. Raphaël. Sharp lower bound on the blow-up rate for critical nonlinear Schrödinger equation. Preprint, 2004. | Zbl

[48] F. Merle and P. Raphaël. Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. to appear. | MR | Zbl

[49] F. Merle and P. Raphaël. Profiles and quantization of the blow-up mass for the critical nonlinear Schrödinger equation. Comm. Math. Phys. to appear. | MR | Zbl

[50] F. Merle and L. Vega. ${L}^{2}$-stability of solitons for KdV equation. Internat. Math. Res. Notices, pages 735-753, 2003. | MR | Zbl

[51] F. Merle and H. Zaag. A Liouville theorem for a vector valued nonlinear heat equation and applications. Math. Ann., 316:103-137, 2000. | MR | Zbl

[52] R. Miura. The Korteweg-de Vries equation : a survey of results. SIAM Rev., 18:412-459, 1976. | MR | Zbl

[53] R. Pego and M. Weinstein. Eigenvalues, and instability of solitary waves. Philos. Trans. Roy. Soc. London Ser. A, 340:47-94, 1992. | MR | Zbl

[54] R. Pego and M. Weinstein. Asymptotic stability of solitary waves. Comm. Math. Phys., 164:305-349, 1994. | MR | Zbl

[55] P. Raphaël. Stability of the $loglog$ bound for blow-up solutions to the critical nonlinear Schrödinger equation. Math. Ann. to appear. | MR | Zbl

[56] J.-C. Saut. Sur quelques généralisations de l'équation de Korteweg-de Vries. J. Math. Pures Appl., 58:21-61, 1979. | MR | Zbl

[57] J.-C. Saut. Remarks on generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J., 42:1011-1026, 1993. | MR | Zbl

[58] P. Schuur. Asymptotic analysis of soliton problems, volume 1232 of Lect. Notes in Math. Springer-Verlag, Berlin, 1986. | MR | Zbl

[59] S. Sulem and P. L. Sulem. The nonlinear Schrödinger equation. Self-focusing and wave collapse, volume 139 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. | MR | Zbl

[60] M. Weinstein. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87:567-576, 1983. | MR | Zbl

[61] M. Weinstein. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Appl. Math., 16:472-491, 1985. | MR | Zbl

[62] M. Weinstein. Lyapunov stability of ground states of nonlinear dispersive equations. Comm. Pure Appl. Math., 39:51-68, 1986. | MR | Zbl

[63] M. Weinstein. On the structure and formation of singularities in solutions to nonlinear dispersive equations. Comm. Partial Differential Equations, 11:545-565, 1986. | MR | Zbl