Stochastic Lagrangian Path for Leray’s 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 have densities $\rho _{s,x}(t,y)$ belonging to $\mathbb {H}^{1,p}_q$ with $p,q\in [1,2)$ and $\frac{3}{p}+\frac{2}{q}>4$:

$$\begin{aligned} \text {d} X_{s,t}=\mathbf{u } (s,X_{s,t})\text {d} t+\sqrt{2\nu }\text {d} W_t,\ \ X_{s,s}=x,\ \ t\geqslant s, \end{aligned}$$

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 $X^n_{s,\cdot }(x)$ 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.

Appendix: Properties of Space \(\widetilde{\mathbb {H}}^{\alpha ,p}_q\)

In this “Appendix” we prove some important properties about the space $\widetilde{\mathbb {H}}^{\alpha ,p}_q$. We need the following lemma, which can be found in [24, p.205] and [33, Lemma 2.2].

Lemma 4.1

• (i) For any $\alpha \in {\mathbb {R}}$ and $p\in (1,\infty )$, there is a $C=C(d,\alpha ,p)>0$ such that

  $$\begin{aligned} \Vert fg\Vert _{\alpha ,p}\leqslant C\Vert f\Vert _{\alpha ,p}\Vert g\Vert _{|\alpha |+1,\infty }. \end{aligned}$$

  (4.1)

• (ii) Let $p\in (1,\infty )$ and $\alpha \in (0,1]$ be fixed. For any $p_1\in [p,\infty )$ and $p_2\in [\frac{p_1}{p_1-1},\infty )$ with $\frac{1}{p}\leqslant \frac{1}{p_1}+\frac{1}{p_2}<\frac{1}{p}+\frac{\alpha }{d}$, there is a constant $C>0$ such that for all $f\in H^{-\alpha ,p_1}$ and $g\in H^{\alpha ,p_2}$,

  $$\begin{aligned} \Vert fg\Vert _{-\alpha ,p}\leqslant C \Vert f\Vert _{-\alpha ,p_1} \Vert g\Vert _{\alpha ,p_2}. \end{aligned}$$

  (4.2)

The following proposition shows that the localized norm enjoys the almost same properties as the global norm $\Vert \cdot \Vert _{\mathbb {H}^{\alpha ,p}_q}$.

Proposition 4.1

Let $p,q\in (1,\infty )$ and $\alpha \in {\mathbb {R}}$.

• (i) For $r\not =r'>0$, there is a constant $C=C(d,\alpha ,r,r')\geqslant 1$ such that for all $f\in \widetilde{\mathbb {H}}^{\alpha ,p}_q$,

  $$\begin{aligned} C^{-1}\sup _{s,z}\Vert f\chi ^{s,z}_{r'}\Vert _{\mathbb {H}^{\alpha ,p}_q}\leqslant \sup _{s,z}\Vert f\chi ^{s,z}_r\Vert _{\mathbb {H}^{\alpha ,p}_q}\leqslant C \sup _{s,z}\Vert f\chi ^{s,z}_{r'}\Vert _{\mathbb {H}^{\alpha ,p}_q}. \end{aligned}$$

  (4.3)

  In other words, the definition of $\widetilde{\mathbb {H}}^{\alpha ,p}_q$ does not depend on the choice of r.

• (ii) Let $(\rho _n)_{n\in {\mathbb {N}}}$ be a family of mollifiers in ${\mathbb {R}}^d$ and $f_n(t,x):=f(t,\cdot )*\rho _n(x)$. For any $f\in \widetilde{\mathbb {H}}^{\alpha ,p}_q$, it holds that $f_n\in L^q_{loc}({\mathbb {R}}; C^\infty _b({\mathbb {R}}^d))$ and for some $C=C(d,\alpha ,p,q)>0$,

  

  (4.4)

  and for any $\varphi \in C^\infty _c({\mathbb {R}}^{d+1})$,

  $$\begin{aligned} \lim _{n\rightarrow \infty }\Vert (f_n-f)\varphi \Vert _{\mathbb {H}^{\alpha ,p}_q}=0. \end{aligned}$$

  (4.5)

• (iii) For any $k\in {\mathbb {N}}$, there is a constant $C=C(d,k,\alpha ,p,q)\geqslant 1$ such that for all $f\in \widetilde{\mathbb {H}}^{\alpha +k,p}_q$,

  

• (iv) Let $p\in (1,\infty )$ and $\alpha \in (0,1]$, $q\in [1,\infty ]$. For any $p_1\in [p,\infty )$ and $p_2\in [\frac{p_1}{p_1-1},\infty )$ with $\frac{1}{p}\leqslant \frac{1}{p_1}+\frac{1}{p_2}<\frac{1}{p}+\frac{\alpha }{d}$, and $\frac{1}{q_1}+\frac{1}{q_2}=\frac{1}{q}$, there is a constant $C>0$ such that

  

• (v) $\mathbb {L}^p_q+\mathbb {L}^\infty _\infty \subsetneq \widetilde{\mathbb {L}}^p_q$.

Proof

• (i) Let $r>r'$. We first prove the right hand side inequality in (4.3). Fix $(s,z)\in {\mathbb {R}}^{d+1}$. Notice that the support of $\chi ^{s,z}_r$ is contained in $Q^{s,z}_{2r}$. Clearly, $Q^{s,z}_{2r}$ can be covered by finitely many $Q_{r'}^{s_i,z_i}, i=1,\cdots , N$, where $N=N(d,r,r')$ does not depend on s, z. Let $(\varphi _i)_{i=1}^{N}$ be the partition of unity associated with $\{Q_{r'}^{s_i,z_i}, i=1,\cdots ,N\}$ so that

  $$\begin{aligned} (\varphi _1+\cdots +\varphi _N)|_{Q^{s,z}_{2r}}=1,\ \mathrm{supp}(\varphi _i)\subset Q_{r'}^{s_i,z_i}. \end{aligned}$$

  Thus, due to $\chi ^{s_i,z_i}_{r'}|_{Q_{r'}^{s_i,z_i}}=1$, by (4.1) we have

  $$\begin{aligned} \Vert f\chi ^{s,z}_{r}\Vert _{\mathbb {H}^{\alpha ,p}_q}&\leqslant \sum _{i=1}^N\Vert f\chi ^{s,z}_{r}\varphi _i\Vert _{\mathbb {H}^{\alpha ,p}_q} =\sum _{i=1}^N\Vert f\chi ^{s_i,z_i}_{r'}\varphi _i\Vert _{\mathbb {H}^{\alpha ,p}_q}\\&\leqslant \sum _{i=1}^N\Vert f\chi ^{s_i,z_i}_{r'}\Vert _{\mathbb {H}^{\alpha ,p}_q}\Vert \varphi _i\Vert _{\mathbb {H}^{|\alpha |+1,\infty }_\infty } \leqslant C\sup _{i=1,\cdots , N}\Vert f\chi ^{s_i,z_i}_{r'}\Vert _{\mathbb {H}^{\alpha ,p}_q}, \end{aligned}$$

  where $C=C(N,\alpha ,d,r,r')>0$, which yields the right hand side inequality in (4.3). On the other hand, since $\chi ^{s,z}_{r'}=\chi ^{s,z}_{2r}\chi ^{s,z}_{r'}$, by what we have proved, we have

  $$\begin{aligned} \Vert f\chi ^{s,z}_{r'}\Vert _{\mathbb {H}^{\alpha ,p}_q}=\Vert f\chi ^{s,z}_{2r}\chi ^{s,z}_{r'}\Vert _{\mathbb {H}^{\alpha ,p}_q} \leqslant C \Vert f\chi ^{s,z}_{2r}\Vert _{\mathbb {H}^{\alpha ,p}_q}\Vert \chi ^{s,z}_{r'}\Vert _{\mathbb {H}^{|\alpha |+1,\infty }_\infty }\leqslant C\Vert f\chi ^{s,z}_{r}\Vert _{\mathbb {H}^{\alpha ,p}_q}, \end{aligned}$$

  where C does not depend on s, z, which gives the left hand side inequality.

• (ii) By the definition of convolutions, it is easy to see that

  $$\begin{aligned} (\chi ^{s,z}_1f_n)(t,x)=\chi ^{s,z}_1(t,x)\cdot (f\chi ^{s,z}_2)(t,\cdot )*\rho _n(x). \end{aligned}$$

  Hence,

  $$\begin{aligned} \Vert \chi ^{s,z}_1f_n\Vert _{\mathbb {H}^{\alpha ,p}_q}\lesssim \Vert \chi ^{s,z}_1\Vert _{\mathbb {H}^{|\alpha |+1,\infty }_\infty }\Vert (f\chi ^{s,z}_2)_n\Vert _{\mathbb {H}^{\alpha ,p}_q} \lesssim \Vert \chi _1\Vert _{\mathbb {H}^{|\alpha |+1,\infty }_\infty }\Vert f\chi ^{s,z}_2\Vert _{\mathbb {H}^{\alpha ,p}_q}, \end{aligned}$$

  which gives (4.4). As for (4.5), it follows by a finitely covering technique.

• (iii) We only prove it for $k=1$. By definition and $\chi ^{s,z}_2\nabla \chi ^{s,z}_1=\nabla \chi ^{s,z}_1$ we have

  $$\begin{aligned} \Vert (\nabla f) \chi ^{s,z}_1\Vert _{\mathbb {H}^{\alpha ,p}_q}&\leqslant \Vert \nabla (f \chi ^{s,z}_1)\Vert _{\mathbb {H}^{\alpha ,p}_q}+\Vert f \nabla \chi ^{s,z}_1\Vert _{\mathbb {H}^{\alpha ,p}_q}\\&\lesssim \Vert f \chi ^{s,z}_1\Vert _{\mathbb {H}^{\alpha +1,p}_q}+\Vert f\chi ^{s,z}_2\Vert _{\mathbb {H}^{\alpha ,p}_q}\Vert \nabla \chi ^{s,z}_1\Vert _{\mathbb {H}^{|\alpha |+1,\infty }_\infty }, \end{aligned}$$

  which in turn gives the right hand side estimate by (i). The left hand side inequality is similar.

• (iv) By (4.2) and $\chi ^{s,z}_2\chi ^{s,z}_1=\chi ^{s,z}_1$, we have

  $$\begin{aligned} \Vert (fg)\chi ^{s,z}_1\Vert _{\mathbb {H}^{-\alpha ,p}_q}=\Vert (f\chi ^{s,z}_2) (g\chi ^{s,z}_1)\Vert _{\mathbb {H}^{-\alpha ,p}_q} \leqslant \Vert f\chi ^{s,z}_2\Vert _{\mathbb {H}^{-\alpha ,p_1}_{q_1}} \Vert g\chi ^{s,z}_1\Vert _{\mathbb {H}^{\alpha ,p_2}_{q_2}}. \end{aligned}$$

  The desired estimate follows by (i).

• (v) Let $\mathbb {Z}^d$ be the set of all lattice points. Define

  $$\begin{aligned} f(t,x):=\mathbf{1}_{[0,1]}(t)\sum _{z\in \mathbb {Z}^d}|x-z|^{-d/p}\mathbf{1}_{|x-z|\leqslant 1}. \end{aligned}$$

  It is easy to see that $f\in \widetilde{\mathbb {L}}^p_q$, but $f\notin \mathbb {L}^p_q+\mathbb {L}^\infty _\infty$. $\quad \square$