Asymptotics and blow-up for mass critical nonlinear dispersive equations

Frank Merle

Chinese Annals of Mathematics, Series B ›› 2017, Vol. 38 ›› Issue (2) : 579 -590.

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Chinese Annals of Mathematics, Series B ›› 2017, Vol. 38 ›› Issue (2) : 579 -590. DOI: 10.1007/s11401-017-1084-7
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Asymptotics and blow-up for mass critical nonlinear dispersive equations

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Abstract

The author considers mass critical nonlinear Schrödinger and Korteweg-de Vries equations. A review on results related to the blow-up of solution of these equations is given.

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Dispersive nonlinear PDE / Criticality / Asymptotics / Blow-up / Global solution / Soliton

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Frank Merle. Asymptotics and blow-up for mass critical nonlinear dispersive equations. Chinese Annals of Mathematics, Series B, 2017, 38(2): 579-590 DOI:10.1007/s11401-017-1084-7

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References

[1]

Berestycki H., Lions P.-L.. Nonlinear scalar field equations I: Existence of a ground state. Arch. Rational Mech. Anal., 1983, 82: 313-345

[2]

Bourgain J.. Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Am. Math. Soc., 1999, 12: 145-171

[3]

Bourgain J., Wang W.. Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. S. Nor. Pisa, 1998, 25: 197-215

[4]

Brezis H., Coron J. M.. Convergence of solutions of H-systems or how to blow bubbles. Arch. Rational Mech. Anal., 1985, 89: 21-56

[5]

Cazenave T., Weissler F.. Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors, 1989, Berlin: Springer 18-29

[6]

Colliander J., Keel M., Staffilani G. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3. Ann. of Math., 2008, 167: 767-865

[7]

Fibich G., Merle F., Raphaël P.. Proof of a spectral property related to the singularity formation for the critical NLS. Phys. D, 2006, 220: 1-13

[8]

Ginibre J., Velo G.. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 1995, 133: 50-68

[9]

Glangetas L., Merle F.. A Geometrical Approach of Existence of Blow-up Solution in H1 for Nonlinear Schrödinger Equations, Publications du Laboratoire d’Analyse Numérique, 1995

[10]

Glassey R.. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys., 1977, 18: 1794-1797

[11]

Kato T.. On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor., 1987, 46: 113-129

[12]

Kenig C.. Recent developments on the global behavior to critical nonlinear dispersive equations. Proceedings of the International Congress of Mathematicians, 2010, New Delhi: Hindustan Book Agency 326-338

[13]

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

[14]

Killip R., Tao T., Visan M.. The cubic nonlinear Schrödinger equation in two dimensions with radial data. J. Eur. Math. Soc., 2009, 11: 1203-1258

[15]

Killip R., Visan M.. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Amer. J. Math., 2010, 132: 361-424

[16]

Krieger J., Nakanishi K., Schlag W.. Global dynamics away from the ground state for the energycritical nonlinear wave equation. Math. Z., 2012, 272: 297-316

[17]

Krieger J., Schlag W.. Non-generic blow-up solutions for the critical focusing NLS in 1-D. Jour. Eur. Math. Soc., 2009, 11: 1-125

[18]

Krieger J., Schlag W., Tataru D.. Renormalization and blow-up for charge one equivariant critical wave maps. Invent. Math., 2008, 171: 543-615

[19]

Krieger J., Schlag W., Tataru D.. Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation. Duke Math. J., 2009, 147: 1-53

[20]

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, 1988, 38: 3837-3843

[21]

Lions P.-L.. The concentration-compactness principle in the calculus of variations: The limit case I and II. Rev. Mat. Ibero., 1985, 1: 45-121

[22]

Martel Y., Merle F.. A Liouville theorem for the critical generalized Korteweg–de Vries equation. J. Math. Pures Appl., 2000, 79: 339-425

[23]

Martel Y., Merle F.. Instability of solitons for the critical generalized Korteweg–de Vries equation. Geom. Funct. Anal., 2001, 11: 74-123

[24]

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., 2002, 155: 235-280

[25]

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., 2002, 15: 617-664

[26]

Martel Y., Merle F.. Nonexistence of blow-up solution with minimal L2-mass for the critical gKdV equation. Duke Math. J., 2002, 115: 385-408

[27]

Martel, Y., Merle, F. and Raphaël, P., Blow-up for critical gKdV equation I: Dynamics near the soliton, Acta Math., to appear. arXiv: 1204.4625

[28]

Martel, Y., Merle, F. and Raphaël, P., Blow-up for critical gKdV equation II: Minimal mass solution, J.E.M.S., to appear.

[29]

Martel, Y., Merle, F. and Raphaël, P., Blow-up for critical gKdV equation III: Exotic regimes, Annali Scuola Norm. Sup. di Pisa, to appear. arXiv:1209.2510

[30]

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

[31]

Merle F.. Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys., 1990, 129: 223-240

[32]

Merle F.. On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math., 1992, 45: 203-254

[33]

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

[34]

Merle F.. Asymptotics for critical nonlinear dispersive equations. Proceedings of the International Congress of Mathematicians, 2014

[35]

Merle F., Raphaël P.. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Func. Anal., 2003, 13: 591-642

[36]

Merle F., Raphaël P.. On universality of blow-up profile for L2 critical nonlinear Schrödinger equation. Invent. Math., 2004, 156: 565-672

[37]

Merle F., Raphaël P.. The blow-up dynamics and upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Ann. of Math., 2005, 161: 157-222

[38]

Merle F., Raphaël P.. Profiles and quantization of the blow-up mass for critical nonlinear Schrödinger equation. Commun. Math. Phys., 2005, 253: 675-704

[39]

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., 2006, 19: 37-90

[40]

Merle F., Raphaël P., Szeftel J.. The instability of Bourgain-Wang solutions for the L2 critical NLS. Amer. Jour. Math., 2013, 135: 967-1017

[41]

Merle F., Raphaël P., Rodnianski I.. Blow-up dynamics for smooth data equivariant solutions to the energy critical Schrödinger map problem. Invent. Math., 2013, 193: 249-365

[42]

Merle, F., Raphaël, P. and Rodnianski, I., Type IIblow up for the energy supercritical NLS, preprint.

[43]

Nakanishi K., Schlag W.. Global dynamics above the ground state energy for the cubic NLS equation in 3D. Arch. Ration. Mech. Anal., 2012, 203: 809-851

[44]

Perelman G.. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré, 2001, 2: 605-673

[45]

Raphaël P.. Stability of the log-log bound for blow-up solutions to the critical nonlinear Schrödinger equation. Math. Ann., 2005, 331: 577-609

[46]

Raphaël P.. Blow up bubbles in Hamiltonian evolution equations: A quantitative approach. Proceedings of the International Congress of Mathematicians, 2014

[47]

Raphaël P., Rodnianski I.. Stable blow-up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems. Publ. Math. Inst. Hautes Etudes Sci., 2012, 115: 1-122

[48]

Rodnianski I., Sterbenz J.. On the formation of singularities in the critical O(3) s-model. Ann. of Math., 2010, 172: 187-242

[49]

Sterbenz J., Tataru D.. Regularity of wave-maps in dimension 2+1. Comm. Math. Phys., 2010, 298: 139-230

[50]

Tao T., Visan M., Zhang X.. Minimal-mass blowup solutions of the mass-critical NLS. Forum Math., 2008, 20: 881-919

[51]

Weinstein M. I.. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 1983, 87: 567-576

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