Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$: Existence and (non-)triviality
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911791501541376 |
|---|---|
| author | Jin, Ruhong Perkowski, Nicolas |
| author_facet | Jin, Ruhong Perkowski, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04911 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$: Existence and (non-)triviality Jin, Ruhong Perkowski, Nicolas Probability Mathematical Physics 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. |
| title | Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$: Existence and (non-)triviality |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2403.04911 |