Exponential Mixing for 2D Stochastic Damped Euler Equation Driven by Bounded Noise

Summary · qwen2.5:32b

Researchers prove exponential mixing for a two-dimensional stochastic damped Euler equation on a torus, showing that with sufficiently large damping, a unique invariant measure exists and the system converges exponentially fast to equilibrium without viscous regularization. This result is novel as it establishes a $W^{1,\infty}$ estimate for vorticity, enabling a compact absorbing mechanism in $C(\mathbb{T}^2)$ and demonstrating that strong linear damping can replace viscosity's role in achieving mixing properties.

Excerpt

arXiv:2606.29352v2 Announce Type: replace Abstract: In this paper, we study the long-time behaviour of the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Unlike stochastic Navier-Stokes or fractionally dissipative Euler equations, the model possesses no viscous regularisation, so the classical parabolic smoothing is unavailable. We prove that when the damping coefficient is sufficiently large, the associated Markov semigroup admits a unique invariant measure and converges exponentially fast to equilibrium. The key ingredient is the establishment of a global-in-time $W^{1,\infty}$ estimate for the vorticity. This estimate yields a compact absorbing mechanism in $C(\mathbb{T}^2)$, which enables us to establish the uniqueness of the invariant measure and exponential mixing. To the best of our knowledge, this is the first exponential mixing result for a genuinely inviscid stochastic Euler-type equation. Our approach demonstrates that sufficiently strong linear damping can effectively replace the compactness mechanism usually provided by viscosity and is expected to be applicable to other inviscid or weakly dissipative stochastic partial differential equations driven by bounded random forcing.