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Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$: Existence and (non-)triviality

Ruhong Jin, Nicolas Perkowski

Abstract

We investigate fractional stochastic Navier-Stokes equations in $d\ge 3$, driven by the random force $(-Δ)^{\fracθ{2}}ξ$ which, as we show, corresponds to a fractional version of the Landau-Lifshitz random force in the physics literature. We obtain the existence and uniqueness of martingale solutions on the torus $\mathbb T^d$ for $θ> \frac{d}{2}$. For $θ\le 1$ the equation is supercritical and we regularize the problem by introducing a Galerkin approximation and we study the large scale behavior of the truncated model on $\RR^d$. We show that the nonlinear term in the Galerkin approximation vanishes on large scales when $θ< 1$ and the model converges to the linearized equation. For $θ= 1$ the nonlinear term gives a nontrivial contribution to the large scale beahvior, and we conjecture that the large scale behavior is given by a linear model with strictly larger effective diffusivity compared to simply dropping the nonlinear term. The effective diffusivity is explicitly given in terms of the model parameters.

Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$: Existence and (non-)triviality

Abstract

We investigate fractional stochastic Navier-Stokes equations in , driven by the random force which, as we show, corresponds to a fractional version of the Landau-Lifshitz random force in the physics literature. We obtain the existence and uniqueness of martingale solutions on the torus for . For the equation is supercritical and we regularize the problem by introducing a Galerkin approximation and we study the large scale behavior of the truncated model on . We show that the nonlinear term in the Galerkin approximation vanishes on large scales when and the model converges to the linearized equation. For the nonlinear term gives a nontrivial contribution to the large scale beahvior, and we conjecture that the large scale behavior is given by a linear model with strictly larger effective diffusivity compared to simply dropping the nonlinear term. The effective diffusivity is explicitly given in terms of the model parameters.
Paper Structure (11 sections, 15 theorems, 128 equations)

This paper contains 11 sections, 15 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Fourier Analysis
  4. Gaussian Analysis
  5. The infinitesimal generator of the truncated equation
  6. Existence and uniqueness of energy solutions for
  7. Solution on the whole space for $\theta \le 1$ and $N<\infty$
  8. Large-scale behavior for
  9. Limiting behaviour
  10. Implication for fluctuating hydrodynamics
  11. Acknowledgement.

Key Result

Theorem 1.1

Let $\theta>\frac{d}{2}$ and $M>0$ and consider an initial distribution $\nu$ with $\frac{d\nu}{d\mu^M} \in L^2(\mu)$, where $\mu^M$ is the divergence-free and mean-free space white noise on $\mathbb{T}^d_M = (\mathbb{R}^d/(M\mathbb{Z}))^d$. There exists a unique-in-law energy solution to equ.SNS wi

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 27 more