## Acta Mathematica

### Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

#### Abstract

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $u_t + (u_{xx} + u^5)_x =0,$which is a canonical mass critical problem, for initial data in H1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L2 norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].

#### Article information

Source
Acta Math., Volume 212, Number 1 (2014), 59-140.

Dates
Received: 13 February 2012
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485801726

Digital Object Identifier
doi:10.1007/s11511-014-0109-2

Mathematical Reviews number (MathSciNet)
MR3179608

Zentralblatt MATH identifier
1301.35137

Rights
2014 © Institut Mittag-Leffler

#### Citation

Martel, Yvan; Merle, Frank; Raphaël, Pierre. Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton. Acta Math. 212 (2014), no. 1, 59--140. doi:10.1007/s11511-014-0109-2. https://projecteuclid.org/euclid.acta/1485801726

#### References

• Bourgain, J. & Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1997), 197–215 (1998).
• Côte R.: Construction of solutions to the L2-critical KdV equation with a given asymptotic behaviour. Duke Math. J., 138, 487–531 (2007)
• Côte, R., Martel, Y. & Merle, F., Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations. Rev. Mat. Iberoam., 27 (2011), 273–302.
• Duyckaerts, T. & Merle, F., Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP, 2008 (2008), Art ID rpn002, 67 pp.
• Duyckaerts T., Merle F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal., 18, 1787–1840 (2009)
• Duyckaerts T., Roudenko S.: Threshold solutions for the focusing 3D cubic Schrödinger equation. Rev. Mat. Iberoam., 26, 1–56 (2010)
• Fibich, G., Merle, F. & Raphaël, P., Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation. Phys. D, 220 (2006), 1–13.
• Hillairet, M. & Raphaël, P., Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation. Anal. PDE, 5 (2012), 777–829.
• Kato, T., On the Cauchy problem for the (generalized) Korteweg–de Vries equation, in Studies in Applied Mathematics, Adv. Math. Suppl. Stud., 8, pp. 93–128. Academic Press, New York, 1983.
• Kenig, C. E. & Merle, F., Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166 (2006), 645–675.
• Kenig, C. E., Ponce, G. & Vega, L., Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46 (1993), 527–620.
• Krieger, J., Nakanishi, K. & Schlag, W., Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Amer. J. Math., 135 (2013), 935–965.
• Krieger J., Schlag W.: Non-generic blow-up solutions for the critical focusing NLS in 1-D. J. Eur. Math. Soc. (JEMS), 11, 1–125 (2009)
• Krieger, J., Schlag, W. & Tataru, D., Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math., 171 (2008), 543–615.
• Landman, M. J., Papanicolaou, G. C., Sulem, C. & Sulem, P.-L., Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A, 38:8 (1988), 3837–3843.
• Martel, Y. & Merle, F., A Liouville theorem for the critical generalized Korteweg–de Vries equation. J. Math. Pures Appl., 79 (2000), 339–425.
• Martel, Y. & Merle, F., Instability of solitons for the critical generalized Korteweg–de Vries equation. Geom. Funct. Anal., 11 (2001), 74–123.
• Martel, Y. & Merle, F., Blow up in finite time and dynamics of blow up solutions for the L2-critical generalized KdV equation. J. Amer. Math. Soc., 15 (2002), 617–664.
• Martel, Y. & Merle, F., Nonexistence of blow-up solution with minimal L2-mass for the critical gKdV equation. Duke Math. J., 115 (2002), 385–408.
• Martel, Y. & Merle, F., Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. of Math., 155 (2002), 235–280.
• Martel Y., Merle F.: Description of two soliton collision for the quartic gKdV equation. Ann. of Math., 174, 757–857 (2011)
• Martel, Y., Merle, F. & Raphaël, P., Blow up for the critical gKdV equation II: minimal mass dynamics. Preprint, 2012.
• Martel, Y., Merle, F. & Raphaël, P., Blow up for the critical gKdV equation III: exotic regimes. To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci..
• Martel, Y., Merle, F. & Tsai, T.-P., Stability in H1 of the sum of K solitary waves for some nonlinear Schrödinger equations. Duke Math. J., 133 (2006), 405–466.
• Merle F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J., 69, 427–454 (1993)
• Merle F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Amer. Math. Soc., 14, 555–578 (2001)
• Merle, F. & Raphaël, P., Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal., 13 (2003), 591–642.
• Merle, F. & Raphaël, P., On universality of blow-up profile for L2 critical nonlinear Schrödinger equation. Invent. Math., 156 (2004), 565–672.
• Merle, F. & Raphaël, P., The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math., 161 (2005), 157–222.
• Merle, F. & Raphaël, P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Comm. Math. Phys., 253 (2005), 675–704.
• Merle, F. & Raphaël, P., On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation. J. Amer. Math. Soc., 19 (2006), 37–90.
• Merle, F., Raphaël, P. & Rodnianski, I., Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem. Invent. Math., 193 (2013), 249–365.
• Merle, F., Raphaël, P. & Szeftel, J., The instability of Bourgain–Wang solutions for the L2. critical NLS. Amer. J. Math., 135 (2013), 967–1017.
• Nakanishi, & K., Schlag, W., Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. J. Differential Equations, 250:5 (2011), 2299–2333.
• Nakanishi, K. & Schlag, W., Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption. Arch. Ration. Mech. Anal., 203 (2012), 809–851.
• Nakanishi, K. & Schlag, W., Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. Partial Differential Equations, 44 (2012), 1–45.
• Perelman G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré, 2, 605–673 (2001)
• Raphaël, P., Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation. Math. Ann., 331 (2005), 577–609.
• Raphaël, P. & Rodnianski, I., Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. Publ. Math. Inst. Hautes Études Sci., 115 (2012), 1–122.
• Raphaël, P. & Schweyer, R., Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow. Comm. Pure Appl. Math., 66 (2013), 414–480.
• Raphaël, P. & Szeftel, J., Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS. J. Amer. Math. Soc., 24 (2011), 471–546.
• Rodnianski, I. & Sterbenz, J., On the formation of singularities in the critical O(3) σ.-model. Ann. of Math., 172 (2010), 187–242.
• Weinstein M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87, 567–576 (1982/83)
• Weinstein M. I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal., 16, 472–491 (1985)