Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise

2024/02/21 by Lü, Huaxiang, Zhu, Xiangchan · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2402.13743

Abstract

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on \mathbbT² \dif u+÷(u⊗ u)\dif t+∇ p\dif tamp;=Δu\dif t + (-Δ)^\fa/2\dif B_t, ÷u=0,driven by derivative of space-time white noise, where \fa∈[0,\frac13). In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution N(0,\frac12(-Δ)^\fa-1). The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.

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