Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

Masaki, Satoshi; Segata, Jun-ichi

doi:10.1016/j.anihpc.2017.04.003

Masaki, Satoshi ; Segata, Jun-ichi

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 283-326.

Résumé

In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical ${\hat{L}}^{r}$ space where ${\hat{L}}^{r} = {f \in S^{'} (R) | {‖ f ‖}_{{\hat{L}}^{r}} = {‖ \hat{f} ‖}_{L^{r^{'}}} < \infty}$ . We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to ${\hat{L}}^{r}$ -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

MR Zbl

DOI : 10.1016/j.anihpc.2017.04.003

Classification : 35Q53, 35B40, 35B30
Mots clés : Generalized Korteweg–de Vries equation, Scattering problem, Threshold solution

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     title = {Existence of a minimal non-scattering solution to the mass-subcritical generalized {Korteweg{\textendash}de} {Vries} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {283--326},
     publisher = {Elsevier},
     volume = {35},
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Masaki, Satoshi; Segata, Jun-ichi. Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 283-326. doi : 10.1016/j.anihpc.2017.04.003. http://www.numdam.org/articles/10.1016/j.anihpc.2017.04.003/

Bibliographie
Cité par

[1] Bégout, P.; Vargas, A. Mass concentration phenomena for the $L^{2}$ -critical nonlinear Schrödinger equation, Trans. Am. Math. Soc., Volume 359 (2007), pp. 5257–5282 | DOI | MR | Zbl

[2] Benjamin, T.B. The stability of solitary waves, Proc. R. Soc. Lond. Ser. A, Volume 328 (1972), pp. 153–183 | MR

[3] Bona, J.L.; Souganidis, P.E.; Strauss, W.A. Stability and instability of solitary waves of Korteweg–de Vries type, Proc. R. Soc. Lond. Ser. A, Volume 411 (1987), pp. 395–412 | MR | Zbl

[4] Bourgain, J. On the restriction and multiplier problems in $R^{3}$ , Geometric Aspects of Functional Analysis (1989–90), Lect. Notes Math., vol. 1469, Springer, Berlin, 1991, pp. 179–191 | DOI | MR | Zbl

[5] Bourgain, J. Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton Math. Ser., Volume vol. 42, Princeton Univ. Press (1995), pp. 83–112 (Princeton, NJ, 1991) | MR | Zbl

[6] Bourgain, J. Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not., Volume 1998 (1998), pp. 253–283 | DOI | MR | Zbl

[7] Boyd, J.P.; Chen, G. Weakly nonlinear wavepackets in the Korteweg–de Vries equation: the KdV/NLS connection, Math. Comput. Simul., Volume 55 (2001), pp. 317–328 | MR | Zbl

[8] Carles, R.; Keraani, S. On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^{2}$ -critical case, Trans. Am. Math. Soc., Volume 359 (2007), pp. 33–62 | MR | Zbl

[9] Cazenave, T. Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, American Mathematical Society, 2003 | DOI | MR | Zbl

[10] Christ, M.; Colliander, J.; Tao, T. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Am. J. Math., Volume 125 (2003), pp. 1235–1293 | DOI | MR | Zbl

[11] Christ, F.M.; Weinstein, M.I. Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal., Volume 100 (1991), pp. 87–109 | DOI | MR | Zbl

[12] Deift, P.; Zhou, X. A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. Math. (2), Volume 137 (1993), pp. 295–368 | DOI | MR | Zbl

[13] Dodson, B. Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., Volume 285 (2015), pp. 1589–1618 | DOI | MR | Zbl

[14] Dodson, B. Global well-posedness and scattering for the defocusing, $L^{2}$ -critical, nonlinear Schrödinger equation when $d = 1$ , Am. J. Math., Volume 138 (2016), pp. 531–569 | DOI | MR | Zbl

[15] Dodson, B. Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Anal. PDE, Volume 3 (2017) no. 5 | MR | Zbl

[16] Grünrock, A. An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., Volume 2004 (2004), pp. 3287–3308 | DOI | MR | Zbl

[17] Grünrock, A. Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., Volume 2005 (2005), pp. 2525–2558 | DOI | MR | Zbl

[18] Grünrock, A.; Vega, L. Local well-posedness for the modified KdV equation in almost critical ${\hat{H}}^{r}$ -spaces, Trans. Am. Math. Soc., Volume 361 (2009), pp. 5681–5694 | DOI | MR | Zbl

[19] Hayashi, N.; Naumkin, P.I. Large time asymptotics of solutions to the generalized Korteweg–de Vries equation, J. Funct. Anal., Volume 159 (1998), pp. 110–136 | DOI | MR | Zbl

[20] Hayashi, N.; Naumkin, P.I. Large time behavior of solutions for the modified Korteweg–de Vries equation, Int. Math. Res. Not., Volume 1999 (1999), pp. 395–418 | DOI | MR | Zbl

[21] Hayashi, N.; Naumkin, P.I. On the modified Korteweg–de Vries equation, Math. Phys. Anal. Geom., Volume 4 (2001), pp. 197–227 | DOI | MR | Zbl

[22] Hayashi, N.; Naumkin, P.I. Final state problem for Korteweg–de Vries type equations, J. Math. Phys., Volume 47 (2006) (16 pp.) | DOI | MR | Zbl

[23] Hyakuna, R.; Tsutsumi, M. On existence of global solutions of Schrödinger equations with subcritical nonlinearity for ${\hat{L}}^{p}$ -initial data, Proc. Am. Math. Soc., Volume 140 (2012), pp. 3905–3920 | DOI | MR | Zbl

[24] Kato, T. On the Cauchy problem for the (generalized) KdV equation, Adv. Math. Suppl. Stud., Stud. Appl. Math., Volume 8 (1983), pp. 93–128 | MR | Zbl

[25] Kato, T. An $L^{q, r}$ -theory for nonlinear Schrödinger equations, Spectral and Scattering Theory and Applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 223–238 | DOI | MR | Zbl

[26] Kenig, C.E.; Merle, F. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., Volume 166 (2006), pp. 645–675 | DOI | MR | Zbl

[27] Kenig, C.E.; Ponce, G.; Vega, L. Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., Volume 40 (1991), pp. 33–69 | DOI | MR | Zbl

[28] Kenig, C.E.; Ponce, G.; Vega, L. Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math., Volume 46 (1993), pp. 527–620 | DOI | MR | Zbl

[29] Kenig, C.E.; Ponce, G.; Vega, L. Nonlinear Wave Equations, Contemp. Math., Volume vol. 263, Amer. Math. Soc., Providence, RI (2000), pp. 131–156 (Providence, RI, 1998) | DOI | MR | Zbl

[30] Killip, R.; Kwon, S.; Shao, S.; Vişan, M. On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., Volume 32 (2012), pp. 191–221 | DOI | MR | Zbl

[31] Killip, R.; Tao, T.; Vişan, M. The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 6, pp. 1203–1258 | MR | Zbl

[32] Killip, R.; Vişan, M. Nonlinear Schrödinger equations at critical regularity, Evolution Equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437 | MR | Zbl

[33] Killip, R.; Vişan, M.; Zhang, X. The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, Volume 1 (2008) no. 2, pp. 229–266 | DOI | MR | Zbl

[34] Koch, H.; Marzuola, J.L. Small data scattering and soliton stability in ${\dot{H}}^{- \frac{1}{6}}$ for the quartic KdV equation, Anal. PDE, Volume 5 (2012), pp. 145–198 | DOI | MR | Zbl

[35] Korteweg, D.J.; de Vries, G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., Volume 39 (1895), pp. 422–443 | DOI | JFM | MR

[36] Lamb, G.L. Jr. Solitons on moving space curves, J. Math. Phys., Volume 18 (1977), pp. 1654–1661 | MR | Zbl

[37] Martel, Y.; Merle, F. Instability of solitons for the critical generalized Korteweg–de Vries equation, Geom. Funct. Anal., Volume 11 (2001), pp. 74–123 | DOI | MR | Zbl

[38] Martel, Y.; Merle, F. Blow up in finite time and dynamics of blow up solutions for the critical generalized KdV equation, J. Am. Math. Soc., Volume 15 (2002), pp. 617–664 | DOI | MR | Zbl

[39] Martel, Y.; Merle, F.; Nakanishi, K.; Raphaël, P. Codimension one threshold manifold for the critical gKdV equation, Commun. Math. Phys., Volume 342 (2016), pp. 1075–1106 | DOI | MR | Zbl

[40] Martel, Y.; Merle, F.; Raphaël, P. Blow up for the critical generalized Korteweg de Vries equation. I: Dynamics near the soliton, Acta Math., Volume 212 (2014), pp. 59–140 | DOI | MR | Zbl

[41] Martel, Y.; Merle, F.; Raphaël, P. Blow up for the critical gKdV equation. II: Minimal mass dynamics, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 1855–1925 | DOI | MR | Zbl

[42] Martel, Y.; Merle, F.; Raphaël, P. Blow up for the critical gKdV equation III: exotic regimes, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 14 (2015), pp. 575–631 | MR | Zbl

[43] Masaki, S. On minimal non-scattering solution for focusing mass-subcritical nonlinear Schrödinger equation, 2013 (preprint available at) | arXiv

[44] Masaki, S. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., Volume 14 (2015), pp. 1481–1531 | MR | Zbl

[45] Masaki, S. Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, 2016 (preprint available at) | arXiv

[46] Masaki, S.; Segata, J. On well-posedness of generalized Korteweg–de Vries equation in scale critical ${\hat{L}}^{r}$ space, Anal. PDE, Volume 9 (2016), pp. 699–725 | DOI | MR

[47] Merle, F.; Vega, L. Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not., Volume 1998 (1998), pp. 399–425 | DOI | MR | Zbl

[48] Molinet, L.; Ribaud, F. Well-posedness results for the generalized Benjamin–Ono equation with small initial data, J. Math. Pures Appl., Volume 83 (2004), pp. 277–311 | DOI | MR | Zbl

[49] Moyua, A.; Vargas, A.; Vega, L. Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not., Volume 1996 (1996), pp. 793–815 | DOI | MR | Zbl

[50] Moyua, A.; Vargas, A.; Vega, L. Restriction theorems and maximal operators related to oscillatory integrals in $R^{3}$ , Duke Math. J., Volume 96 (1999) no. 3, pp. 547–574 | DOI | MR | Zbl

[51] Ponce, G.; Vega, L. Nonlinear small data scattering for the generalized Korteweg–de Vries equation, J. Funct. Anal., Volume 90 (1990), pp. 445–457 | DOI | MR | Zbl

[52] Rammaha, M.A. On the asymptotic behavior of solutions of generalized Korteweg–de Vries equations, J. Math. Anal. Appl., Volume 140 (1989), pp. 228–240 | DOI | MR | Zbl

[53] Schindler, I.; Tintarev, K. An abstract version of the concentration compactness principle, Rev. Mat. Complut., Volume 15 (2002), pp. 417–436 | DOI | MR | Zbl

[54] Schneider, G. Approximation of the Korteweg–de Vries equation by the nonlinear Schrödinger equation, J. Differ. Equ., Volume 147 (1998), pp. 333–354 | DOI | MR | Zbl

[55] Shao, S. The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality, Anal. PDE, Volume 2 (2009), pp. 83–117 | DOI | MR | Zbl

[56] Sidi, A.; Sulem, C.; Sulem, P.L. On the long time behavior of a generalized KdV equation, Acta Appl. Math., Volume 7 (1986), pp. 35–47 | DOI | MR | Zbl

[57] Strauss, W.A. Nonlinear scattering theory at low energy, J. Funct. Anal., Volume 41 (1981), pp. 110–133 | DOI | MR | Zbl

[58] Tao, T. Scattering for the quartic generalised Korteweg–de Vries equation, J. Differ. Equ., Volume 232 (2007), pp. 623–651 | MR | Zbl

[59] Tao, T. Two remarks on the generalised Korteweg–de Vries equation, Discrete Contin. Dyn. Syst., Volume 18 (2007), pp. 1–14 | MR | Zbl

[60] Tomas, P.A. A restriction theorem for the Fourier transform, Bull. Am. Math. Soc., Volume 81 (1975), pp. 477–478 | DOI | MR | Zbl

[61] Vargas, A.; Vega, L. Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite $L^{2}$ norm, J. Math. Pures Appl. (9), Volume 80 (2001), pp. 1029–1044 | DOI | MR | Zbl

[62] Weinstein, M.I. Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1982/83), pp. 567–576 | MR | Zbl

[63] Weinstein, M.I. Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., Volume 39 (1986), pp. 51–67 | DOI | MR | Zbl

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