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.
Keywords: explosion en temps fini, EDP hamiltonienne, KdV
Mot clés : blow-up solutions, hamiltonian PDE, KdV
@incollection{SB_2003-2004__46__219_0, author = {Tzvetkov, Nikolay}, title = {On the long time behavior of {KdV} type equations}, booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937}, series = {Ast\'erisque}, note = {talk:933}, pages = {219--248}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {299}, year = {2005}, mrnumber = {2167208}, zbl = {1074.35079}, language = {en}, url = {http://www.numdam.org/item/SB_2003-2004__46__219_0/} }
TY - CHAP AU - Tzvetkov, Nikolay TI - On the long time behavior of KdV type equations BT - Séminaire Bourbaki : volume 2003/2004, exposés 924-937 AU - Collectif T3 - Astérisque N1 - talk:933 PY - 2005 SP - 219 EP - 248 IS - 299 PB - Association des amis de Nicolas Bourbaki, Société mathématique de France PP - Paris UR - http://www.numdam.org/item/SB_2003-2004__46__219_0/ LA - en ID - SB_2003-2004__46__219_0 ER -
%0 Book Section %A Tzvetkov, Nikolay %T On the long time behavior of KdV type equations %B Séminaire Bourbaki : volume 2003/2004, exposés 924-937 %A Collectif %S Astérisque %Z talk:933 %D 2005 %P 219-248 %N 299 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://www.numdam.org/item/SB_2003-2004__46__219_0/ %G en %F SB_2003-2004__46__219_0
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://www.numdam.org/item/SB_2003-2004__46__219_0/
[1] Solitons and the inverse scattering transform. SIAM, Philadelphia, 1981. | MR | Zbl
and .[2] Blow-up for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Boston, 1995. | MR | Zbl
.[3] Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 29:559-592, 1967. | Zbl
.[4] The stability of solitary waves. Proc. London Math. Soc. (3), 328:153-183, 1972. | MR
.[5] Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A, 272:47-78, 1972. | MR | Zbl
, , and .[6] The stability of solitary waves. Proc. London Math. Soc. (3), 344:363-374, 1975. | MR | Zbl
.[7] The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A, 278:555-601, 1975. | MR | Zbl
and .[8] Stability and instability of solitary waves of Korteweg-de Vries type. Proc. London Math. Soc. (3), 411:395-412, 1987. | MR | Zbl
, , and .[9] Similarity solutions of the generalized Korteweg-de Vries equation. Math. Proc. Cambridge Philos. Soc., 127:323-351, 1999. | MR | Zbl
and .[10] Non existence of -compact solutions of the Kadomtsev-Petviashvili II equation. Math. Ann., 328:525-544, 2004. | MR
and .[11] Global solutions of nonlinear Schrödinger equations, volume 46 of AMS Colloquium Publications. American Mathematical Society, Providence, R.I., 1999. | MR | Zbl
.[12] 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] Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes. 2003. | MR | Zbl
.[14] Orbital stability of standing waves for some nonlinear Schrödinger equation. Comm. Math. Phys., 85:549-561, 1982. | MR | Zbl
and .[15] 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] The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions. Math. Methods Appl. Sci., 5:97-116, 1983. | MR | Zbl
and .[17] 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] Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Preprint, 2003. | MR | Zbl
.[19] 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
, , and .[20] Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 14:187-209, 1997. | Numdam | MR | Zbl
, , and .[21] Uniqueness of solutions for the generalized Korteweg-de Vries equation. SIAM J. Appl. Math., 20:1388-1425, 1989. | MR | Zbl
and .[22] A geometric approach of existence of blow-up solutions. Preprint, 1995.
and .[23] Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal., 74:160-197, 1987. | MR | Zbl
, , and .[24] The analysis of linear partial differential operators I. Springer-Verlag, 1983. | Zbl
.[25] 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] On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett., 10:879-895, 2003. | MR | Zbl
and .[27] 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
, , and .[28] On the concentration of blow-up solutions for the generalized KdV equation critical in . volume 263 of Contemp. Math., pages 131-156. American Mathematical Society, 2000. | MR | Zbl
, , and .[29] Elements of soliton theory. John Wiley & Sons, New York, 1980. | MR | Zbl
[30] Smoothness and exponential decay of -compact solutions of the generalized KdV equation. Comm. Partial Differential Equations, 28:2093-2107, 2003. | MR | Zbl
and .[31] On the stability of KdV multi-solitons. Comm. Pure Appl. Math., 46:867-901, 1993. | MR | Zbl
and .[32] Multi-soliton-type solutions of the generalized KdV equations. Amer. J. Math. to appear. | MR | Zbl
.[33] Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal., 11:74-123, 2001. | MR | Zbl
and .[34] Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rational Mech. Anal., 157:219-254, 2001. | MR | Zbl
and .[35] A Liouville Theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl., 79:339-425, 2000. | MR | Zbl
and .[36] 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
and .[37] Blow-up in finite time and dynamics of blow-up solutions for the -critical generalized KdV equation. J. Amer. Math. Soc., 15:617-663, 2002. | MR | Zbl
and .[38] Nonexistence of blow-up solution with minimal -mass for the critical GKdV. Duke Math. J., 115:385-408, 2002. | MR | Zbl
and .[39] Asymptotic stability of solitons for subcritical generalized KdV equations revisited. Preprint, 2004. | MR | Zbl
and .[40] Stability and asymptotic stability in the energy space of the sum of solitons for subcritical gKdV equations. Comm. Math. Phys., 231:347-373, 2002. | MR | Zbl
, , and .[41] 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] Asymptotics for -minimal blow-up solutions of critical nonlinear Schrödinger equation. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13:553-565, 1996. | Numdam | MR | Zbl
.[43] 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] 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] Sharp upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Geom. Funct. Anal., 13:591-642, 2003. | MR | Zbl
and .[46] On universality of blow-up profile for -critical nonlinear Schrödinger equation. Invent. Math., 156:565-672, 2004. | MR | Zbl
and .[47] Sharp lower bound on the blow-up rate for critical nonlinear Schrödinger equation. Preprint, 2004. | Zbl
and .[48] Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. to appear. | MR | Zbl
and .[49] Profiles and quantization of the blow-up mass for the critical nonlinear Schrödinger equation. Comm. Math. Phys. to appear. | MR | Zbl
and .[50] -stability of solitons for KdV equation. Internat. Math. Res. Notices, pages 735-753, 2003. | MR | Zbl
and .[51] A Liouville theorem for a vector valued nonlinear heat equation and applications. Math. Ann., 316:103-137, 2000. | MR | Zbl
and .[52] The Korteweg-de Vries equation : a survey of results. SIAM Rev., 18:412-459, 1976. | MR | Zbl
.[53] Eigenvalues, and instability of solitary waves. Philos. Trans. Roy. Soc. London Ser. A, 340:47-94, 1992. | MR | Zbl
and .[54] Asymptotic stability of solitary waves. Comm. Math. Phys., 164:305-349, 1994. | MR | Zbl
and .[55] Stability of the bound for blow-up solutions to the critical nonlinear Schrödinger equation. Math. Ann. to appear. | MR | Zbl
.[56] Sur quelques généralisations de l'équation de Korteweg-de Vries. J. Math. Pures Appl., 58:21-61, 1979. | MR | Zbl
.[57] Remarks on generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J., 42:1011-1026, 1993. | MR | Zbl
.[58] Asymptotic analysis of soliton problems, volume 1232 of Lect. Notes in Math. Springer-Verlag, Berlin, 1986. | MR | Zbl
.[59] The nonlinear Schrödinger equation. Self-focusing and wave collapse, volume 139 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. | MR | Zbl
and .[60] Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87:567-576, 1983. | MR | Zbl
.[61] Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Appl. Math., 16:472-491, 1985. | MR | Zbl
.[62] Lyapunov stability of ground states of nonlinear dispersive equations. Comm. Pure Appl. Math., 39:51-68, 1986. | MR | Zbl
.[63] On the structure and formation of singularities in solutions to nonlinear dispersive equations. Comm. Partial Differential Equations, 11:545-565, 1986. | MR | Zbl
.