Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 995–1058 | Cite as

Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

Article

Abstract

We construct a solution for the Complex Ginzburg–Landau equation in a critical case which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows us to prove the stability of the constructed solution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aranson I.S., Kramer L.: The world of the complex Ginzburg–Landau equation. Rev. Modern Phys. 74(1), 99–143 (2002)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Bricmont J., Kupiainen A.: Universality in blow-up for nonlinear heat equations. Nonlinearity 7(2), 539–575 (1994)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Cazenave, T.: Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003Google Scholar
  4. 4.
    Côte R., Zaag H.: Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension. Commun. Pure Appl. Math. 66(10), 1541–1581 (2013)CrossRefMATHGoogle Scholar
  5. 5.
    Ebde M.A., Zaag H.: Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term. SeMA J. 55, 5–21 (2011)CrossRefMATHGoogle Scholar
  6. 6.
    Giga Y., Kohn R.V.: Nondegeneracy of blowup for semilinear heat equations. Commun. Pure Appl. Math. 42(6), 845–884 (1989)CrossRefMATHGoogle Scholar
  7. 7.
    Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg–Landau equation. Differ. Equ. Asymptot. Anal. Math. Phys. 100, 138–152 (1996)MATHGoogle Scholar
  8. 8.
    Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg–Landau equation. II. Contraction methods. Commun. Math. Phys. 187(1), 45–79 (1997)ADSCrossRefMATHGoogle Scholar
  9. 9.
    Hocking L.M., Stewartson K.: On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. Lond. Ser. A 326, 289–313 (1972)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Hocking L.M., Stewartson K., Stuart J.T., Brown S.N.: A nonlinear instability in plane parallel flow. J. Fluid Mech. 51, 705–735 (1972)ADSCrossRefMATHGoogle Scholar
  11. 11.
    Kolodner, P., Bensimon, D., Surko, M.: Traveling wave convection in an annulus. Phys. Rev. Lett. 60, 1723–1988Google Scholar
  12. 12.
    Kolodner P., Slimani S., Aubry N., Lima R.: Characterization of dispersive chaos and related states of binary-fluid convection. Phys. D 85(1-2), 165–224 (1995)CrossRefMATHGoogle Scholar
  13. 13.
    Merle F.: Solution of a nonlinear heat equation with arbitrary given blow-up points. Commun. Pure Appl. Math. 45(3), 263–300 (1992)CrossRefMATHGoogle Scholar
  14. 14.
    Merle F., Zaag H.: Stability of the blow-up profile for equations of the type \({u_t={\Delta} u+\vert u\vert ^{p-1}u}\). Duke Math. J. 86(1), 143–195 (1997)CrossRefMATHGoogle Scholar
  15. 15.
    Masmoudi N., Zaag H.: Blow-up profile for the complex Ginzburg–Landau equation. J. Funct. Anal. 225, 1613–1666 (2008)CrossRefMATHGoogle Scholar
  16. 16.
    Nouaili N., Zaag H.: A liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up. Trans. Am. Math. Soc. 362(7), 3391–3434 (2010)CrossRefMATHGoogle Scholar
  17. 17.
    Nguyen, V.T., Zaag, H.: Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, 2015 (Submitted 2015)Google Scholar
  18. 18.
    Nouaili, N., Zaag, H.: Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation. Commun. Partial Differ. Equ. (7), 1197–1217, 2015Google Scholar
  19. 19.
    Plecháč P., Šverák V.: On self-similar singular solutions of the complex Ginzburg–Landau equation. Commun. Pure Appl. Math. 54(10), 1215–1242 (2001)CrossRefMATHGoogle Scholar
  20. 20.
    Popp S., Stiller O., Kuznetsov E., Kramer L.: The cubic complex Ginzburg–Landau equation for a backward bifurcation. Phys. D 114(1–2), 81–107 (1998)CrossRefMATHGoogle Scholar
  21. 21.
    Rottschäfer V.: Multi-bump, self-similar, blow-up solutions of the Ginzburg–Landau equation. Phys. D 237(4), 510–539 (2008)CrossRefMATHGoogle Scholar
  22. 22.
    Rottschäfer V.: Asymptotic analysis of a new type of multi-bump, self-similar, blowup solutions of the Ginzburg–Landau equation. Eur. J. Appl. Math. 24(1), 103–129 (2013)CrossRefMATHGoogle Scholar
  23. 23.
    Raphaël P., Schweyer R.: Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow. Commun. Pure Appl. Math. 66(3), 414–480 (2013)CrossRefMATHGoogle Scholar
  24. 24.
    Raphaël P., Schweyer R.: On the stability of critical chemotactic aggregation. Math. Ann. 359(1–2), 267–377 (2014)CrossRefMATHGoogle Scholar
  25. 25.
    Stewartson K., Stuart J.T.: A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545 (1971)ADSCrossRefMATHGoogle Scholar
  26. 26.
    Tayachi, S., Zaag, H.: Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term. 2015. arXiv preprint: arXiv:1506.08306, 2015
  27. 27.
    Velázquez J.J.L.: Higher-dimensional blow up for semilinear parabolic equations. Commun. Partial Differ. Equ. 17(9-10), 1567–1596 (1992)CrossRefMATHGoogle Scholar
  28. 28.
    Velázquez J.J.L.: Classification of singularities for blowing up solutions in higher dimensions. Trans. Am. Math. Soc. 338(1), 441–464 (1993)CrossRefMATHGoogle Scholar
  29. 29.
    Velázquez J.J.L.: Estimates on the (n−1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J. 42(2), 445–476 (1993)CrossRefMATHGoogle Scholar
  30. 30.
    Velázquez, J.J.L., Galaktionov, V.A., Herrero, M.A.: The space structure near a blow-up point for semilinear heat equations: a formal approach. Zh. Vychisl. Mat. i Mat. Fiz. 31(3), 1991Google Scholar
  31. 31.
    Zaag H.: Blow-up results for vector-valued nonlinear heat equations with no gradient structure. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(5), 581– (1998)ADSCrossRefMATHGoogle Scholar
  32. 32.
    Zaag H.: A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure. Commun. Pure Appl. Math. 54(1), 107–133 (2001)CrossRefMATHGoogle Scholar
  33. 33.
    Zaag H.: On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(5), 505–542 (2002)ADSCrossRefMATHGoogle Scholar
  34. 34.
    Zaag H.: One-dimensional behavior of singular N-dimensional solutions of semilinear heat equations. Commun. Math. Phys. 225(3), 523–549 (2002)ADSCrossRefMATHGoogle Scholar
  35. 35.
    Zaag, H.: Regularity of the blow-up set and singular behavior for semilinear heat equations. In Mathematics & Mathematics Education (Bethlehem, 2000), pp. 337–347. World Science Publishing, River Edge, 2002Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CEREMADE, UMR 7534Université Paris IX-Dauphine, PSL Research UniversityParisFrance
  2. 2.LAGA, CNRS (UMR 7539)Université Paris 13, Sorbonne Paris CitéVilletaneuseFrance

Personalised recommendations