Preprint
Nonlinear Instability in the 2D Kuramoto-Sivashinsky equation
arXiv (Cornell University)
22 Jul 2026
Abstract
In this paper we analyze the Kuramoto-Sivashinsky equation (KSE), a model of flame-front propagation, on a two-dimensional square torus of arbitrary size2 LwithL > π . In this case, the linearized equation at the origin admits a finite number of growing modes, which corresponds to the positive eigenvalues of the linear operator- Δ² - Δ . The problem of analyzing the long-time behavior of solutions of the 2D KSE in two spatial dimensions remains largely open; the only global existence results are for sufficiently small tori, or for sufficiently anisotropic and thin domains, due to the lack of good a priori estimates. The main purpose of this paper is to analyze the instability around growing modes at the nonlinear level. More precisely, we consider the maximal growing modeλ₀and we show that there is a finite dimensional manifold of initial data of sizeεarbitrarily small such that the corresponding solutions become of sizeO(1)over a time-scale of order\rm log(ε⁻ ¹) . The proof is based on several ingredients such as a sharp quantitative construction of an approximate solution bifurcating from the maximal linearly growing mode, a fixed point argument with exponential weights to construct local in time solutions, and a continuation argument based on sharp energy estimates and para-differential calculus.
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Details
- Title
- Nonlinear Instability in the 2D Kuramoto-Sivashinsky equation
- Creators
- David M Ambrose - Drexel UniversityAnna L Mazzucato - Pennsylvania State UniversityRiccardo Montalto - University of Milan
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022197291104721