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Global Solutions of the Two-Dimensional Kuramoto–Sivashinsky Equation with a Linearly Growing Mode in Each Direction

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Abstract

We consider the Kuramoto–Sivashinsky equation in two space dimensions. We establish the first proof of global existence of solutions in the presence of a linearly growing mode in both spatial directions for sufficiently small data. We develop a new method to this end, categorizing wavenumbers as low (linearly growing modes), intermediate (linearly decaying modes that serve as energy sinks for the low modes), and high (strongly linearly decaying modes). The low and intermediate modes are controlled by means of a Lyapunov function, while the high modes are controlled with operator estimates in function spaces based on the Wiener algebra.

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Acknowledgements

The first author is grateful to the National Science Foundation for support through grant DMS-1907684. The second author is grateful to the National Science Foundation for support through grant DMS-1909103.

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Correspondence to David M. Ambrose.

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Communicated by Changpin Li.

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Ambrose, D.M., Mazzucato, A.L. Global Solutions of the Two-Dimensional Kuramoto–Sivashinsky Equation with a Linearly Growing Mode in Each Direction. J Nonlinear Sci 31, 96 (2021). https://doi.org/10.1007/s00332-021-09748-8

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  • DOI: https://doi.org/10.1007/s00332-021-09748-8

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