Tohoku Mathematical Journal

A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting

Tadahiro Oh and Oana Pocovnicu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this note, we prove almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on $\mathbb{T}^d$, $d = 3, 4,$ and $5$, with random initial data below the energy space.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 3 (2017), 455-481.

Dates
First available in Project Euclid: 12 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1505181626

Digital Object Identifier
doi:10.2748/tmj/1505181626

Mathematical Reviews number (MathSciNet)
MR3695994

Zentralblatt MATH identifier
06814879

Subjects
Primary: 35L05: Wave equation
Secondary: 35L71: Semilinear second-order hyperbolic equations

Keywords
nonlinear wave equation probabilistic well-posedness almost sure global existence finite speed of propagation

Citation

Oh, Tadahiro; Pocovnicu, Oana. A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting. Tohoku Math. J. (2) 69 (2017), no. 3, 455--481. doi:10.2748/tmj/1505181626. https://projecteuclid.org/euclid.tmj/1505181626


Export citation

References

  • H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175.
  • H. Bahouri and J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 6, 783–789.
  • Á. Bényi and T. Oh, The Sobolev inequality on the torus revisited, Publ. Math. Debrecen, 83 (2013), no. 3, 359–374.
  • Á. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursions in Harmonic Analysis, Vol. 4, 3–25, Appl. Numer. Harmon. Anal., Birkhüser/Springer, Cham, 2015.
  • Á. Bényi, T. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^d$, $d\geq 3$, Trans. Amer. Math. Soc. Ser. B 2 (2015), 1–50.
  • J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445.
  • N. Burq, L. Thomann and N. Tzvetkov, Global infinite energy solutions for the cubic wave equation, Bull. Soc. Math. France. 143 (2015), no. 2, 301–313.
  • N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475.
  • N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. 16 (2014), no. 1, 1–30.
  • M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations. [math.AP].
  • J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\mathbb{T})$, Duke Math. J. 161 (2012), no. 3, 367–414.
  • J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, Jour. Func. Anal. 110 (1992), 96–130.
  • M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. 132 (1990), 485–509.
  • M. Grillakis, Regularity for the wave equation with a critical nonlinearity, Commun. Pure Appl. Math. 45 (1992), 749–774.
  • L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Reprint of the second (1990) edition. Classics in Mathematics. Springer-Verlag, Berlin, 2003. x+440 pp.
  • L. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Letters 1 (1994), 211–223.
  • M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980.
  • H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357–426.
  • J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbb{R}^3$, Comm. Partial Differential Equations 39 (2014), no. 12, 2262–2283.
  • J. Lührmann and D. Mendelson, On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $\mathbb{R}^3$, New York J. Math. 22 (2016), 209–227.
  • K. Nakanishi, Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity, Comm. Partial Differential Equations 24 (1999), no. 1–2, 185–221.
  • K. Nakanishi, Scattering Theory for Nonlinear Klein-Gordon Equation with Sobolev Critical Power, Internat. Math. Res. Not. 1 (1999), 31–60.
  • T. Oh and O. Pocovnicu, Probabilistic well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$, J. Math. Pures Appl. (9) 105 (2016), no. 3, 342–366.
  • T. Oh and J. Quastel, On Cameron-Martin theorem and almost sure global existence, Proc. Edinb. Math. Soc. 59 (2016), 483–501.
  • O. Pocovnicu, Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d = 4$ and 5, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2521–2575.
  • M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York-London, 1975. xv+361 pp.
  • J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), no. 3, 503–518.
  • J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Inter. Math. Research Not. 7 (1994), 303–309.
  • M. Struwe, Globally regular solutions to the $u^5$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1988), 495–513.
  • C. Sun and B. Xia, Probabilistic well-posedness for supercritical wave equation on $\mathbb{T}^3$, arXiv:1508.00228 [math.AP].
  • T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dyn. Partial Differ. Equ. 3 (2006), no. 2, 93–110.
  • N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100.