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Pré-Publication, Document De Travail Année : 2023

Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction

Résumé

We consider systems of damped wave equations with a state-dependent damping coefficient and perturbed by a Gaussian multiplicative noise. Initially, we investigate their well-posedness, under quite general conditions on the friction. Subsequently, we study the validity of the so-called Smoluchowski-Kramers diffusion approximation. We show that, under more stringent conditions on the friction, in the small-mass limit the solution of the system of stochastic damped wave equations converges to the solution of a system of stochastic quasi-linear parabolic equations. In this convergence, an additional drift emerges as a result of the interaction between the noise and the state-dependent friction. The identification of this limit is achieved by using a suitable generalization of the classical method of perturbed test functions, tailored to the current infinite dimensional setting.

Dates et versions

hal-04383799 , version 1 (09-01-2024)

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Sandra Cerrai, Arnaud Debussche. Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction. 2024. ⟨hal-04383799⟩
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