Abstract
In this paper, we establish the Freidlin-Wentzell type large deviation principle for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $\Delta (|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.
Highlights
We are interested in the asymptotic behaviour of porous media equations with small multiplicative noise
We are concerned with the following porous media equations with stochastic forcing du(t, x) = ∆(|u(t, x)|m−1u(t, x))dt + Φ(u(t, x))dW (t) in TN × (0, T ], u(·, 0) = u0(·) ∈ Lm+1(TN ) on TN, (1.1)
The purpose of this paper is to prove that the kinetic solution to the stochastic porous mediumtype equations (1.2) satisfies Freidlin-Wentzell type large deviation principle (LDP) in the space L1([0, T ]; L1(TN )), which is a more delicate result compared with [29] and [23]
Summary
We are interested in the asymptotic behaviour of porous media equations with small multiplicative noise. The purpose of this paper is to prove that the kinetic solution to the stochastic porous mediumtype equations (1.2) satisfies Freidlin-Wentzell type LDP in the space L1([0, T ]; L1(TN )), which is a more delicate result compared with [29] and [23]. The present paper is the first work towards establishing the LDP directly for the kinetic solution to the stochastic porous medium-type equations (1.2). Due to the fact that the kinetic solutions are living in a rather irregular space comparing to various type solutions for parabolic SPDEs, it is a challenge to establish LDP for the stochastic porous media equations with general noise force.
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