Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection

Abstract

We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on the two-dimensional torus with and without advection by an incompressible vector field. We prove local existence of mild solutions for arbitrary data in L². We then study the issue of global existence. We prove global existence for the KSE in the presence of advection for arbitrary data, provided the advecting velocity field v satisfies certain conditions that ensure the dissipation time of the associated hyperdiffusion-advection equation is sufficiently small. In the absence of advection, global existence can be shown only if the linearized operator does not admit any growing mode and for sufficiently small initial data.

Keywords:

• Two dimension
• Kuramoto-Sivashinsky
• mixing
• global existence
• mild solutions
• enhanced diffusion
• diffusion time

2010 Mathematics Subject Classification:

• 35K25
• 35K58
• 76E06
• 76F25

Acknowledgments

The authors thank Gautam Iyer for useful discussions. A.M. was partially supported by the US National Science Foundation grants DMS-1909103 and DMS-1615457. Part of this work was conducted while the second author was on leave from Penn State University to New York University-Abu Dhabi.

Additional information

Funding

The authors thank Gautam Iyer for useful discussions. A.M. was partially supported by the US National Science Foundation grants DMS-1909103 and DMS-1615457. Part of this work was conducted while the second author was on leave from Penn State University to New York University-Abu Dhabi.