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Nondispersive solutions to the L 2-critical Half-Wave Equation

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Abstract

We consider the focusing L 2-critical half-wave equation in one space dimension,

$$i \partial_t u = D u - |u|^2 u$$

, where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold \({M_{*} > 0}\) such that all H 1/2 solutions with \({\|u\|_{L^2} < M_*}\) extend globally in time, while solutions with \({\|u\|_{L^2} \geq M_*}\) may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass \({\|u_0\|_{L^2} = M_*}\). More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E 0 > 0 and the linear momentum \({P_0 \in \mathbb{R}}\). In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L 2-critical nonlinear PDEs with nonlocal dispersion.

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References

  1. Banica V., Carles R., Duyckaerts T.: Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation. Commun. Partial Differ. Equ. 36(3), 487–531 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourgain, J., Wang, W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1997)(1–2), 197–215 (1998) (Dedicated to Ennio De Giorgi)

  3. Burq N., Gérard P., Tzvetkov N.: Two singular dynamics of the nonlinear Schrödinger equation on a plane domain. Geom. Funct. Anal. 13(1), 1–19 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, D., Majda, A.J., McLaughlin, D.W., Tabak, E.G.: Dispersive wave turbulence in one dimension. Phys. D 152/153, 551–572 (2001) Advances in nonlinear mathematics and science

    Google Scholar 

  5. Calderón A.-P.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. U.S.A. 53, 1092–1099 (1965)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Cazenave, T.: Semilinear Schrödinger equations. Courant Lecutre Nores in Mathematics 10 (2003), New York University Courant Institute of Mathematical

  7. Dodson, B.: Global well-posedness and scattering for the mass critical nonlinear Schrdinger equation with mass below the mass of the ground state. Preprint available at: arXiv:1104.1114

  8. Duyckaerts T., Merle F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18(6), 1787–1840 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Elgart A., Schlein B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60(4), 500–545 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frank, R.L., Lenzmann, E.: Uniqueness of ground states for fractional Laplacians in R. Acta Math. Preprint available at arXiv:1009.4042 (to appear, 2012)

  11. Frank, R.L., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Preprint available at arXiv:1302.2652 (2013)

    Google Scholar 

  12. Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: Effective dynamics for boson stars. Nonlinearity 20(5), 1031–1075 (2007)

    Google Scholar 

  13. Fröhlich J., Jonsson B.L.G., Lenzmann E.: Boson stars as solitary waves. Commun. Math. Phys. 274(1), 1–30 (2007)

    Article  ADS  MATH  Google Scholar 

  14. Fröhlich J., Lenzmann E.: Blowup for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60(11), 1691–1705 (2007)

    Article  MATH  Google Scholar 

  15. Gérard, P., Grellier, S.: The cubic Szegő equation. Ann. Sci. éc. Norm. Supér. (4) 43(5), 761–810 (2010)

    Google Scholar 

  16. Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)

    MathSciNet  ADS  MATH  Google Scholar 

  17. Judovič, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)

  18. 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. 166(3), 645–675 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Killip R., Tao T., Visan M.: The cubic nonlinear Schrödinger equation in two dimensions with radial data. J. Eur. Math. Soc. (JEMS) 11(6), 1203–1258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Killip, R., Visan, M., Zhang, X.: The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE 1(2), 229–266 (2008)

    Google Scholar 

  21. Kirkpatrick K., Lenzmann E., Staffilani G.: On the continuum limit for discrete NLS with long-range interactions. Commun. Math. Phys. 317(3), 563–591 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Krieger J., Martel Y., Raphaël P.: Two-soliton solutions to the three-dimensional gravitational Hartree equation. Commun. Pure Appl. Math. 62(11), 1501–1550 (2009)

    Article  MATH  Google Scholar 

  24. Kwong, M.K.: Uniqueness of positive solutions of Δu − u + u p = 0 in R n. Arch. Ration. Mech. Anal. 105(3), 243–266 (1989)

  25. Majda A.J., McLaughlin D.W., Tabak E.G.: A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci. 7(1), 9–44 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Martel Y.: Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Am. J. Math. 127(5), 1103–1140 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Martel Y., Merle F.: Nonexistence of blow-up solution with minimal L 2-mass for the critical gKdV equation. Duke Math. J. 115(2), 385–408 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Martel, Y., Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(6), 849–864 (2006)

    Google Scholar 

  29. Merle F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power.. Duke Math J. 69(2), 427–454 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Merle F., Raphael P.: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13(3), 591–642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Merle, F.: Nonexistence of minimal blow-up solutions of equations iu t  = −Δu−k(x)| u|4/N u in R N. Ann. Inst. H. Poincaré Phys. Théor. 64(1), 33–85 (1996)

  32. Merle, F.: Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Commun. Math. Phys. 129(2), 223–240 (1990)

    Google Scholar 

  33. Merle, F., Martel, Y., Raphaël, P.: Blow up for the critical gKdV I: dynamics near the soliton (preprint, 2012)

  34. Merle, F., Martel, Y., Raphaël, P.: Blow up for the critical gKdV II: the minimal mass solution (preprint, 2012)

  35. Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. (2) 161(1), 157–222 (2005)

    Google Scholar 

  36. Merle, F., Raphael, P.: On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19(1), 37–90 (electronic) (2006)

    Google Scholar 

  37. Merle, F., Raphaël, P., Szeftel, J.: Stable self-similar blow-up dynamics for slightly L 2 super-critical NLS equations. Geom. Funct. Anal. 20(4), 1028–1071 (2010)

    Google Scholar 

  38. Ogawa T.: A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations. Nonlinear Anal. 14(9), 765–769 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pocovnicu O.: Explicit formula for the solution of the Szegö equation on the real line and applications. Discret. Contin. Dyn. Syst. 31(3), 607–649 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Raphaël P., Szeftel J.: Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS. J. Am. Math. Soc. 24(2), 471–546 (2011)

    Article  MATH  Google Scholar 

  41. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton (with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) (1993)

  42. Tartar, L.: An introduction to Sobolev spaces and interpolation spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3. Springer, Berlin (2007)

  43. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87(4), 567–576 (1982/83)

    Google Scholar 

  44. Weinstein M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, EPFL, Lausanne, 1051, Switzerland

    Joachim Krieger

  2. Mathematisches Institut, Universität Basel, Rheinsprung 21, Basel, 4051, Switzerland

    Enno Lenzmann

  3. Laboratoire J.A. Dieudonné, UMR CNRS 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, NICE Cedex 02, France

    Pierre Raphaël

Authors
  1. Joachim Krieger
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  2. Enno Lenzmann
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  3. Pierre Raphaël
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Correspondence to Pierre Raphaël.

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Communicated by S. Müller

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Krieger, J., Lenzmann, E. & Raphaël, P. Nondispersive solutions to the L 2-critical Half-Wave Equation. Arch Rational Mech Anal 209, 61–129 (2013). https://doi.org/10.1007/s00205-013-0620-1

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