2021/02/09 by David M. Ambrose, Anna L. Mazzucato, Ambrose, David M. +1 · 1 citation
Computer Science · Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2102.05093
openalex publication_date 2021/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In two spatial dimensions, there are very few global existence results for\nthe Kuramoto-Sivashinsky equation. The majority of the few results in the\nliterature are strongly anisotropic, i.e. are results of thin-domain type. In\nthe spatially periodic case, the dynamics of the Kuramoto-Sivashinsky equation\nare in part governed by the size of the domain, as this determines how many\nlinearly growing Fourier modes are present. The strongly anisotropic results\nallow linearly growing Fourier modes in only one of the spatial directions. We\nprovide here the first proof of global solutions for the two-dimensional\nKuramoto-Sivashinsky equation with a linearly growing mode in both spatial\ndirections. We develop a new method to this end, categorizing wavenumbers as\nlow (linearly growing modes), intermediate (linearly decaying modes which serve\nas energy sinks for the low modes), and high (strongly linearly decaying\nmodes). The low and intermediate modes are controlled by means of a Lyapunov\nfunction, while the high modes are controlled with operator estimates in\nfunction spaces based on the Wiener algebra.\n