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Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

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Abstract

We construct a minimal mass blow up solution of the modified Benjamin–Ono equation (mBO)

$$\begin{aligned} u_{t}+(u^3-D^1 u)_{x}=0, \end{aligned}$$
(mBO)

which is a standard mass critical dispersive model. Let \(Q\in H^{\frac{1}{2}}\), \(Q>0\), be the unique ground state solution of \(D^1 Q +Q=Q^3\), constructed using variational arguments by Weinstein (Commun. Part. Differ. Equations 12:1133–1173, 1987a; J. Differ. Equations 69:192–203, 1987b) and Albert et al. (Proc. R. Soc. Lond. A 453:1233–1260, 1997), and whose uniqueness was recently proved by Frank and Lenzmann (Acta Math. 210:261–318, 2013). We show the existence of a solution S of (mBO) satisfying \(\Vert S \Vert _{L^2}=\Vert Q\Vert _{L^2}\) and

$$\begin{aligned} S(t)-\frac{1}{\lambda ^{\frac{1}{2}}(t)} Q\left( \frac{\cdot - x(t)}{\lambda (t)}\right) \rightarrow 0\quad \text{ in } \ H^{\frac{1}{2}}(\mathbb R) \text{ as } \ t\downarrow 0, \end{aligned}$$

where

$$\begin{aligned} \lambda (t)\sim t,\quad x(t) \sim -|\ln t| \quad \hbox {and}\quad \Vert S(t)\Vert _{\dot{H}^{\frac{1}{2}}} \sim t^{-\frac{1}{2}}\Vert Q\Vert _{\dot{H}^{\frac{1}{2}}} \quad \hbox {as}\ t\downarrow 0. \end{aligned}$$

This existence result is analogous to the one obtained by Martel et al. (J. Eur. Math. Soc. 17:1855–1925, 2015) for the mass critical generalized Korteweg-de Vries equation. However, in contrast with the (gKdV) equation, for which the blow up problem is now well-understood in a neighborhood of the ground state, S is the first example of blow up solution for (mBO). The proof involves the construction of a blow up profile, energy estimates as well as refined localization arguments, developed in the context of Benjamin–Ono type equations by Kenig et al. (Ann. Inst. H. Poincaré Anal. Non Lin. 28:853–887, 2011). Due to the lack of information on the (mBO) flow around the ground state, the energy estimates have to be considerably sharpened in the present paper.

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Notes

  1. Amick and Toland proved the following stronger statement in [4]: any nonconstant bounded solution of (2.2) is either \(Q_\mathrm{BO}\) (up to translation) or a periodic wave solution.

  2. Which follows by using that \(\int _{\mathbb R}\mathcal {H}(Q')xQ'dx=0\), since \(\mathcal {H}(x\phi )=x\mathcal {H}\phi \) if \(\int \phi dx=0\).

References

  1. Abdelouhab, L., Bona, J.L., Felland, M., Saut, J.-C.: Nonlocal models for nonlinear, dispersive waves. Phys. D 40, 360–392 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albert, J., Bona, J.L., Saut, J.-C.: Model equations for waves in stratified fluids. Proc. R. Soc. Lond. A 453, 1233–1260 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amick, C.J., Toland, J.F.: Uniqueness of Benjamin’s solitary-wave solution of the Benjamin–Ono equation. IMA J. Appl. Math. 46, 21–28 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amick, C.J., Toland, J.F.: Uniqueness and related analytic properties of the Benjamin–Ono equation–a nonlinear Neumann problem in the plane. Acta. Math. 167, 107–126 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Angulo, J., Bona, J.L., Linares, F., Scialom, M.: Scaling, stability and singularities for nonlinear, dispersive wave equations: the critical case. Nonlinearity 15, 759–786 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bajvsank, B., Coifman, R.: On singular integrals. Proc. Symp. Pure Math, pp. 1–17. AMS, Providence RI (1966)

  7. Banica, V.: Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3, 139–170 (2004)

  8. Banica, V., Carles, R., Duyckaerts, T.: Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation. Commun. Part. Differ. Equations 36, 487–531 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Benjamin, T.B.: Internal waves of permanent form in fluids of great depth. J. Fluid. Mech. 29, 559–592 (1967)

    Article  MATH  Google Scholar 

  10. Bona, J.L., Kalisch, H.: Singularity formation in the generalized Benjamin–Ono equation. Disc. Contin. Dyn. Syst. 11, 27–45 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond. 411, 395–412 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bourgain, J., Wang, W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25(4), 197–215 (1997)

  13. Calderon, A.-P.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. USA 53, 1092–1099 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cazenave, T.: Semilinear Schrödinger equations. New York University, Courant Institute, New York (2003)

    Book  MATH  Google Scholar 

  15. Coifman, R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. Combet, V.: Multi-soliton solutions for the supercritical gKdV equations. Commun. Part. Differ. Equations 36, 380–419 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Combet, V., Genoud, F.: Classification of minimal mass blow-up solutions for an \(L^2\) critical inhomogeneous NLS. To appear in J. Evol. Equ. (2015). arXiv:1503.08915 (preprint)

  18. Combet, V., Martel, Y.: Sharp asymptotics for the minimal mass blow up solution of critical gKdV equation (2016). arXiv:1602.03519 (preprint)

  19. Côte, R., Martel, Y., Merle, F.: Construction of multi-soliton solutions for the \(L^2\)-supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27, 273–302 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cui, S., Kenig, C.E.: Weak continuity of the flow map for the Benjamin–Ono equation on the line. J. Fourier Anal. Appl. 16, 1021–1052 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dawson, L., McGahagan, H., Ponce, G.: On the decay properties of solutions to a class of Schrödinger equations. Proc. Am. Math. Soc. 136, 2081–2090 (2008)

    Article  MATH  Google Scholar 

  22. Frank, R., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R}\). Acta Math. 210, 261–318 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hmidi, T., Keraani, S.: Blowup theory for the critical nonlinear Schrödinger equations revisited. Int. Math. Res. Not. 46, 2815–2828 (2005)

    Article  MATH  Google Scholar 

  24. Kato, T.: On the Cauchy problem for the (generalized) Korteweg–de Vries equation. Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, pp. 93–128. Academic Press, New York (1983)

  25. Kenig, C.E., Martel, Y.: Asymptotic stability of solitons for the Benjamin–Ono equation. Rev. Mat. Iberoam. 25, 909–970 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kenig, C.E., Martel, Y., Robbiano, L.: Local well-posedness and blow-up in the energy space for a class of \(L^2\) critical dispersion generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré, Anal. Non Lin. 28, 853–887 (2011)

  27. Kenig, C.E., Ponce, G., Vega, L.: On the generalized Benjamin–Ono equation. Trans. Am. Math. Soc. 342, 155–172 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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. 46, 527–620 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kenig, C.E., Takaoka, H.: Global well-posedness of the modified Benjamin–Ono equation with initial data in \(H^{\frac{1}{2}}\). Int. Math. Res. Not. 1–44 (2006) (Art. ID: 95702)

  30. Klein, C., Peter, R.: Numerical study of blow-up in solutions to generalized Kadomtsev–Petviashvili equations. Discrete Contin. Dyn. Syst. Ser. B 19, 1689–1717 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Klein, C., Saut, J.-C.: A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation. Phys. D 295(296), 46–65 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krieger, J., Lenzmann, E., Raphaël, P.: Nondispersive solutions to the \(L^2\)-critical half-wave equation. Arch. Ration. Mech. Anal. 209, 61–129 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Le Coz, S., Martel, Y., Raphaël, P.: Minimal mass blow up solutions for a double power nonlinear Schrödinger equation. To appear in Rev. Mat. Iberoam (2014). arXiv:1406.6002 (preprint)

  34. Linares, F., Pilod, D., Saut, J.-C.: Dispersive perturbations of Burgers and hyperbolic equations I: local theory. SIAM J. Math. Anal. 46, 1505–1537 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Martel, Y., Merle, F.: Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. Math. 155, 235–280 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Martel, Y., Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18, 55–80 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Martel, Y., Merle, F., Nakanishi, K., Raphaël, P.: Codimension one threshold manifold for the critical gKdV equation. Commun. Math. Phys. 342, 1075–1106 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Martel, Y., Merle, F., Raphaël, P.: Blow-up for the critical generalized Korteweg–de Vries equation I: dynamics near the soliton. Acta Math. 212, 59–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Martel, Y., Merle, F., Raphaël, P.: Blow-up for the critical generalized Korteweg–de Vries equation II: minimal mass dynamics. J. Eur. Math. Soc. 17, 1855–1925 (2015)

    Article  MATH  Google Scholar 

  43. Martel, Y., Merle, F., Raphaël, P.: Blow-up for the critical generalized Korteweg–de Vries equation III: exotic regimes, vol. XIV, pp. 575–631. Ann. Sc. Norm. Sup. Pisa (2015)

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Merle, F.: Nonexistence of minimal blow-up solutions of equations \(iu_t = -\Delta u -k(x)|u|^{\frac{4}{N}}u\) in \(\mathbb{R}^N\). Ann. IHP Phys. Théor. 64, 33–85 (1996)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Merle, F., Raphaël, P.: On universality of blow-up profile for \(L^2\) critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. Merle, F., Tsutsumi, Y.: \(L^2\) concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity. J. Differ. Equations 84, 205–214 (1990)

    Article  MATH  Google Scholar 

  51. Molinet, L., Ribaud, F.: Well-posedness results for the generalized Benjamin–Ono equation with arbitrary large initial data. Int. Math. Res. Not. 70, 3757–3795 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  52. Molinet, L., Ribaud, F.: Well-posedness results for the generalized Benjamin–Ono equation with small initial data. J. Math. Pures Appl. 83, 277–311 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  53. Ono, H.: Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39, 1082–1091 (1975)

    Article  MATH  Google Scholar 

  54. Perelman, G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Inst. Henri Poincaré 2, 605–673 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  56. Tao, T.: Global well-posedness of the Benjamin–Ono in \(H^1(\mathbb{R})\). J. Hyperbolic Differ. Equations 1, 27–49 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  57. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1982/1983)

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Weinstein, M.I.: On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Commun. Part. Differ. Equations 11, 545–565 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  60. Weinstein, M.I.: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Commun. Part. Differ. Equations 12, 1133–1173 (1987a)

    Article  MathSciNet  MATH  Google Scholar 

  61. Weinstein, M.I.: Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities. J. Differ. Equations 69, 192–203 (1987b)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank Carlos Kenig for drawing their attention to the blow up problem for (mBO) and Jean-Claude Saut for encouraging and helpful discussions. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000 while Y.M. was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2015 semester. D.P. would like to thank the École polytechnique for the kind hospitality during the elaboration of part of this work. This work was also partially supported by CNPq/Brazil, Grant 302632/2013-1 and by the Project ERC 291214 BLOWDISOL.

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  1. CMLS, École Polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France

    Yvan Martel

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, CEP: 21945-970, Brazil

    Didier Pilod

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  1. Yvan Martel
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Correspondence to Yvan Martel.

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Communicated by Nalini Anantharaman.

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Martel, Y., Pilod, D. Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation. Math. Ann. 369, 153–245 (2017). https://doi.org/10.1007/s00208-016-1497-8

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