Abstract
The Kuramoto–Sivashinsky equation is a nonlinear parabolic partial differential equation, which describes the instability and turbulence of waves in chemical reactions and laminar flames. The aim of this work is to prove the large deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise. To establish the large deviation principle, the weak convergence approach is used, which relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation.
Highlights
The theory of large deviations studies the exponential decay of probabilities in certain random systems
Large deviation theory for small noise stochastic differential equation (SDE) has been extensively studied, which was introduced by Friedlin and Wentzell [18], who established the large deviations for such SDEs driven by finitely many Brownian motions
The proofs of large deviation principle (LDP) using the aforesaid theory have relied on first approximating the original problem by timediscretization so that LDP can be shown on the resulting simpler problems via contraction principle and showing that LDP holds in the limit using exponential probability estimates that are specific to the model under study
Summary
The theory of large deviations studies the exponential decay of probabilities in certain random systems. A class of nonlocal stochastic Kuramoto–Sivashinsky (SKS) equations driven by Poisson random measures, and the existence and uniqueness of weak solution were studied in [3]. The sufficient conditions are obtained by proving certain qualitative properties such as existence, uniqueness and tightness of the analogous controlled processes and convergence to its limiting zero noise equation, which leads to a simple, short and more straightforward proof as compared to the other methods. The Lp(D) norm is denoted by h(t, ·) Lp for a function h(t, x) with respect to the variable x ∈ D
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