Global Existence and Analyticity for the 2D Kuramoto–Sivashinsky Equation

  • David M. Ambrose
  • Anna L. Mazzucato


There is little analytical theory for the behavior of solutions of the Kuramoto–Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the linearized behavior depends on the size of the torus—in particular, for different sizes of the domain, there are different numbers of linearly growing modes. We prove that small solutions exist for all time if there are no linearly growing modes, proving also in this case that the radius of analyticity of solutions grows linearly in time. In the general case (i.e., in the presence of a finite number of growing modes), we make estimates for how the radius of analyticity of solutions changes in time.


Two dimension Kuramoto–Sivashinsky Radius of analyticity Global existence Mild solutions Wiener algebra 

Mathematics Subject Classification

35K25 35K58 35B65 35B10 



The authors thank Edriss Titi for helpful conversations. The authors are also grateful to the National Science Foundation for support through NSF Grants DMS-1515849 (to Ambrose) and DMS-1615457 (to Mazzucato). The authors acknowledge the hospitality and support of the Institute for Computational and Experimental Research in Mathematics (ICERM) during the Semester Program on “Singularities and Waves In Incompressible Fluids”, where part of this work was discussed. ICERM receives major funding from NSF and Brown University.


  1. 1.
    Ambrose, D.M.: Small strong solutions for time-dependent mean field games with local coupling. C. R. Math. Acad. Sci. Paris 354(6), 589–594 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrose, D.M.: Strong solutions for time-dependent mean field games with non-separable Hamiltonians. J. Math. Pures Appl. (2018).
  3. 3.
    Benachour, S., Kukavica, I., Rusin, W., Ziane, M.: Anisotropic estimates for the two-dimensional Kuramoto–Sivashinsky equation. J. Dyn. Differ. Equ. 26(3), 461–476 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biswas, A., Jolly, M.S., Martinez, V.R., Titi, E.S.: Dissipation length scale estimates for turbulent flows: a Wiener algebra approach. J. Nonlinear Sci. 24(3), 441–471 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biswas, A., Swanson, D.: Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in \(\mathbb{R}^n\). J. Differ. Equ. 240(1), 145–163 (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Doering, Charles R., Titi, Edriss S.: Exponential decay rate of the power spectrum for solutions of the Navier–Stokes equations. Phys. Fluids 7(6), 1384–1390 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duchon, J., Robert, R.: Global vortex sheet solutions of Euler equations in the plane. J. Differ. Equ. 73(2), 215–224 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Giacomelli, L., Otto, F.: New bounds for the Kuramoto–Sivashinsky equation. Commun. Pure Appl. Math. 58(3), 297–318 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goodman, J.: Stability of the Kuramoto–Sivashinsky and related systems. Commun. Pure Appl. Math. 47(3), 293–306 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grujić, Zoran, Kukavica, Igor: Space analyticity for the Navier–Stokes and related equations with initial data in \(L^p\). J. Funct. Anal. 152(2), 447–466 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ioakim, X., Smyrlis, Y.-S.: Analyticity for Kuramoto–Sivashinsky-type equations in two spatial dimensions. Math. Methods Appl. Sci. 39(8), 2159–2178 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Milgrom, T., Ambrose, D.M.: Temporal boundary value problems in interfacial fluid dynamics. Appl. Anal. 92(5), 922–948 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Molinet, L.: A bounded global absorbing set for the Burgers–Sivashinsky equation in space dimension two. C. R. Acad. Sci. Paris Sér. I Math. 330(7), 635–640 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Molinet, L.: Local dissipativity in \(L^2\) for the Kuramoto–Sivashinsky equation in spatial dimension 2. J. Dyn. Differ. Equ. 12(3), 533–556 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of the Kuramoto–Sivashinsky equations: nonlinear stability and attractors. Physica D 16(2), 155–183 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oliver, Marcel, Titi, Edriss S.: Remark on the rate of decay of higher order derivatives for solutions to the Navier–Stokes equations in \({ R}^n\). J. Funct. Anal. 172(1), 1–18 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Oliver, Marcel, Titi, Edriss S.: On the domain of analyticity of solutions of second order analytic nonlinear differential equations. J. Differ. Equ. 174(1), 55–74 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  20. 20.
    Sell, G.R., Taboada, M.: Local dissipativity and attractors for the Kuramoto–Sivashinsky equation in thin \(2{\rm D}\) domains. Nonlinear Anal. 18(7), 671–687 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stanislavova, M., Stefanov, A.: The Kuramoto–Sivashinsky equation in \(\text{R}^1\) and \(\text{ R }^2\): effective estimates of the high-frequency tails and higher Sobolev norms. arXiv:0711.4005, November 2007
  22. 22.
    Tadmor, E.: The well-posedness of the Kuramoto–Sivashinsky equation. SIAM J. Math. Anal. 17(4), 884–893 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Taylor, M.E.: Partial Differential Equations III Nonlinear Equations. Applied Mathematical Sciences, vol. 117, 2nd edn. Springer, New York (2011)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsPenn State UniversityUniversity ParkUSA

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