Nonlinear Instability in the 2D Kuramoto-Sivashinsky equation

In this paper we analyze the Kuramoto-Sivashinsky equation (KSE), a model of flame-front propagation, on a two-dimensional square torus of arbitrary size $2 L$ with $L > \pi$. 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 $- \Delta^2 - \Delta$. 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 mode $\lambda_0$ and we show that there is a finite dimensional manifold of initial data of size $\varepsilon$ arbitrarily small such that the corresponding solutions become of size $O(1)$ over a time-scale of order ${\rm log}(\varepsilon^{- 1})$. 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.