Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations

Abstract

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution $\mathbf{u}$ of 3D-NSE and each $(s,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^3$, we show the existence of weak solutions to the following SDE, which has a density ${\rho }_{s,x}(t,y)$ belonging to ${\mathbb{H}}_q^{1,p}$ provided $p,q\in [1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$\mathrm{d}{X}_{s,t}=\mathbf{u}(s,X_{s,t})\mathrm{d}t+\sqrt{2\nu }\mathrm{d}{W}_t,X_{s,s}=x,t\ge s,$$ where $W$ is a three dimensional standard Brownian motion, $\nu >0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X_{s,\cdot }^n(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf{u}}_n$ weakly converges to $X_{s,\cdot }(x)$ so that $X$ is a Markov process in almost sure sense.

25pages

References in corpus (5)

• Noise Prevents Singularities in Linear Transport Equations
• Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations
• Heat kernel and ergodicity of SDEs with distributional drifts
• Regular Flows for Diffusions with Rough Drifts
• Brownian motion with general drift

Cited by in corpus (15)

• Well-posedness of distribution dependent SDEs with singular drifts
• Taming singular stochastic differential equations: A numerical method
• SDEs with critical time dependent drifts: strong solutions
• LDP and CLT for SPDEs with Transport Noise
• $L^q(L^p)$-theory of stochastic differential equations
• SDEs with critical time dependent drifts: weak solutions
• Stochastic Lagrangian Flows for SDEs with rough coefficients
• On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$
• Solution theory of fractional SDEs in complete subcritical regimes
• Regularity properties of jump diffusions with irregular coefficients
• Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations
• SDEs with random and irregular coefficients
• Maximum principle for non-uniformly parabolic equations and applications
• Multidimensional Stable Driven McKean-Vlasov SDEs with Distributional Interaction Kernel: Critical Thresholds and Related Models
• Expansion and attraction of RDS: long time behavior of the solution to singular SDE