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

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Abstract

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.

Keywords

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

Mathematics Subject Classification

35K25 35K58 35B65 35B10 

Notes

Acknowledgements

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.

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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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