Stationary solutions to the compressible Navier–Stokes system driven by stochastic forces

Abstract

We study the long-time behavior of solutions to a stochastically driven Navier–Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue–Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential tool in order to obtain the global-in-time estimate is the stationarity of solutions on each approximation level, which provides a certain regularizing effect. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory, due to the underlying martingale structure of the noise.

Appendix A: Auxiliary results

In this final section, we collect several auxiliary results concerning the two notions of stationarity introduced in Definitions 2.7 and 2.8. First of all, we observe that it is actually enough to consider Definition 2.8 for $q=1$.

Lemma A.1

Let $k\in \mathbb {N}_0$, $p,q\in [1,\infty )$. If $\mathbf {U}$ is stationary on $L^1_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ in the sense of Definition 2.8 and $\mathbf {U}\in L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$$\mathbb {P}$-a.s. then $\mathbf {U}$ is stationary on $L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$.

Proof

According to the assumption, for all $f\in C_b(L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$, it holds

$$\begin{aligned} \mathbb {E}[f(\mathbf {U})]=\mathbb {E}[f(\mathcal {S}_\tau \mathbf {U})]. \end{aligned}$$

If $f\in C_b(L^q_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$ then for all $R\in \mathbb {N}$

$$\begin{aligned} \mathbf {U}\mapsto f(\mathbf {U}\,\mathbf {1}_{|\mathbf {U}|\le R})\in C_b(L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))) \end{aligned}$$

hence

$$\begin{aligned} \mathbb {E}[f(\mathbf {U}\,\mathbf {1}_{|\mathbf {U}|\le R})]=\mathbb {E}[f((\mathcal {S}_\tau \mathbf {U})\mathbf {1}_{|\mathcal {S}_\tau \mathbf {U}|\le R})]. \end{aligned}$$

Finally, since $\mathbf {U}\in L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$$\mathbb {P}$-a.s., we obtain that

$$\begin{aligned} \mathbf {U}\,\mathbf {1}_{|\mathbf {U}|\le R}\rightarrow \mathbf {U}\quad \text {in}\quad L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))\quad \mathbb {P}\text {-a.s.} \end{aligned}$$

and we conclude by the dominated convergence. $\square$

Next, we show that for the case of stochastic processes with continuous trajectories, the two definitions are equivalent.

Lemma A.2

Let $k\in \mathbb {N}_0$, $p\in [1,\infty )$. An $W^{k,p}(\mathbb {T}^3)$-valued measurable stochastic process $\mathbf {U}$ with $\mathbb {P}$-a.s. continuous trajectories is stationary on $W^{k,p}(\mathbb {T}^3)$ in the sense of Definition 2.7 if and only if it is stationary on $L^1_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ in the sense of Definition 2.8.

Proof

Let us first show that Definition 2.8 implies Definition 2.7. Let $\tau \ge 0$ and $t_1,\dots ,t_n\in [0,\infty )$. Let $(\psi _m)$ be a smooth and compactly supported approximation to the identity on $\mathbb {R}$ and define

$$\begin{aligned} \Psi _m(\mathbf {U})= \left( \int _0^\infty \mathbf {U}(s)\psi _m(t_1-s)\mathrm {d}s,\dots , \int _0^\infty \mathbf {U}(s)\psi _m(t_n-s)\mathrm {d}s\right) . \end{aligned}$$

If $\varphi \in C_b([W^{k,p}(\mathbb {T}^3)]^n)$ then $\varphi \circ \Psi _m\in C_b(L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$ and therefore

$$\begin{aligned} \mathbb {E}[\varphi \circ \Psi _m(\mathcal {S}_\tau \mathbf {U})]=\mathbb {E}[\varphi \circ \Psi _m(\mathbf {U})]. \end{aligned}$$

Sending $m\rightarrow \infty$ we obtain due to the continuity of $\mathbf {U}$ and the dominated convergence theorem that

$$\begin{aligned} \mathbb {E}[\varphi (\mathbf {U}(t_1+\tau ),\dots ,\mathbf {U}(t_n+\tau ))]=\mathbb {E}[\varphi (\mathbf {U}(t_1),\dots ,\mathbf {U}(t_n))] \end{aligned}$$

and the claim follows.

To show the converse implication, let us fix $\tau \ge 0$ and an equidistant partition $0=t_1<\cdots<t_n<\cdots <\infty$ with mesh size $\Delta t=\frac{\tau }{m}$ for some $m\in \mathbb {N}$. Observe that there is an one-to-one correspondence between sequences ${\hat{\mathbf {U}}}_m=(\mathbf {U}(t_1),\mathbf {U}(t_2),\dots )\in \ell ^1_{\mathrm{loc}}(W^{k,p}(\mathbb {T}^3))$ and piecewise constant functions in $L^1_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ given by ${\tilde{\mathbf {U}}}_m(t)=\mathbf {U}(t_i)$ if $t\in [t_i,t_{i+1})$. Moreover, it is an isometry in the following sense

$$\begin{aligned} \sum _{i=1}^N\Vert {\hat{\mathbf {U}}}_m(t_i)\Vert _{W^{k,p}(\mathbb {T}^3)}=\int _0^{N\Delta t}\Vert {\tilde{\mathbf {U}}}_m(t)\Vert _{W^{k,p}(\mathbb {T}^3)}\,\mathrm {d}t. \end{aligned}$$

Thus, if $\Phi$ denotes this isometry and $\varphi \in C_b (L^1_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$, then $\varphi \circ \Phi \in C_b(\ell ^1_{\mathrm{loc}}(W^{k,p}(\mathbb {T}^3)))$. Consequently,

$$\begin{aligned} \mathbb {E}[\varphi ({\tilde{\mathbf {U}}}_m)]=\mathbb {E}[\varphi (\mathcal {S}_\tau {\tilde{\mathbf {U}}}_m)] \end{aligned}$$

follows from Definition 2.7. Due to the continuity of $\mathbf {U}$ we may send $m\rightarrow \infty$ which completes the proof. $\square$

The following result proves that weak continuity together with a uniform bound is enough for the equivalence of Definitions 2.7 and 2.8 to hold true.

Corollary A.3

The statement of Lemma A.2 remains valid if the trajectories of $\mathbf {U}$ are $\mathbb {P}$-a.s. weakly continuous and for all $T>0$

$$\begin{aligned} \sup _{t\in [0,T]}\Vert \mathbf {U}\Vert _{W^{k,p}(\mathbb {T}^3)}<\infty \quad \mathbb {P}\text {-a.s.} \end{aligned}$$

(A.1)

Proof

Let $(\varphi _\varepsilon )$ be an approximation to the identity on $\mathbb {T}^3$. Since $\mathbf {U}$ has weakly continuous trajectories in $W^{k,p}(\mathbb {T}^3)$ and satisfies (A.1), the process $\mathbf {U}^\varepsilon :=\mathbf {U}*\varphi _\varepsilon$ has strongly continuous trajectories in $W^{k,p}(\mathbb {T}^3)$. Hence the equivalence of the two notions of stationarity from Lemma A.2 holds.

Now, let $\mathbf {U}$ be stationary on $L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ in the sense of Definition 2.8. That is, for every $f\in C_b(L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$ we have

$$\begin{aligned} \mathbb {E}[f(\mathcal {S}_\tau \mathbf {U})]=\mathbb {E}[f(\mathbf {U})]. \end{aligned}$$

Since $\mathbf {U}\mapsto f(\mathbf {U}*\varphi _\varepsilon )$ also belongs to $C_b(L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$ we deduce that

$$\begin{aligned} \mathbb {E}[f(\mathbf {U}^\varepsilon )]=\mathbb {E}[f([\mathcal {S}_\tau \mathbf {U}]*\varphi _\varepsilon )]=\mathbb {E}[f(\mathcal {S}_\tau \mathbf {U}^\varepsilon )]. \end{aligned}$$

So, $\mathbf {U}^\varepsilon$ is stationary in the sense of Definition 2.8 and due to Lemma A.2, $\mathbf {U}^\varepsilon$ is stationary in the sense of Definition 2.7. In addition, $\mathbf {U}^\varepsilon (t)\rightarrow \mathbf {U}(t)$ strongly in $W^{k,p}(\mathbb {T}^3)$ for every $t\in [0,\infty )$. Therefore, if $g\in C_b([W^{k,p}(\mathbb {T}^3)]^n)$, we may use dominated convergence in order to pass to the limit in expressions of the form

$$\begin{aligned} \mathbb {E}[g(\mathbf {U}^\varepsilon (t_1),\dots , \mathbf {U}^\varepsilon (t_n))]=\mathbb {E}[g(\mathbf {U}^\varepsilon (t_1+\tau ),\dots , \mathbf {U}^\varepsilon (t_n+\tau ))]. \end{aligned}$$

Stationarity of $\mathbf {U}$ in the sense of Definition 2.7 follows.

To show the converse implication, assume that $\mathbf {U}$ is stationary in the sense of Definition 2.7. By the same argument as above, it follows that $\mathbf {U}^\varepsilon$ is stationary in the sense of Definition 2.7 hence stationary in the sense of Definition 2.8. In other words, for every $f\in C_b(L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$,

$$\begin{aligned} \mathbb {E}[f(\mathbf {U}^\varepsilon )]=\mathbb {E}[f(\mathcal {S}_\tau \mathbf {U}^\varepsilon )]. \end{aligned}$$

According to (A.1) we obtain that $\mathbf {U}^\varepsilon \rightarrow \mathbf {U}$ in $L^1_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ and the dominated convergence theorem yields the claim. $\square$

As the next step, we show that both notions of stationarity introduced in Definitions 2.7 and 2.8 are stable under weak convergence.

Lemma A.4

Let $k\in \mathbb {N}_0, p,q\in [1,\infty )$ and let $(\mathbf {U}_m)$ be a sequence of random variables taking values in $L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$. If, for all $m\in \mathbb {N}$, $\mathbf {U}_m$ is stationary on $L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ in the sense of Definition 2.8 and

$$\begin{aligned} \mathbf {U}_m\rightharpoonup \mathbf {U}\quad \text {in}\quad L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))\quad {\mathbb {P}}\text {-a.s.,} \end{aligned}$$

then $\mathbf {U}$ is stationary on $L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$.

Proof

Stationarity of $\mathbf {U}_m$ implies that for every $f\in C_b(L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$ and every $\tau \ge 0$

$$\begin{aligned} \mathbb {E}[ f(\mathcal {S}_\tau \mathbf {U}_m)]=\mathbb {E}[ f(\mathbf {U}_m)]. \end{aligned}$$

(A.2)

Moreover, it follows from the above weak convergence and the weak continuity of

$$\begin{aligned} \mathcal {S}_\tau :L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))\rightarrow L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))) \end{aligned}$$

that for every $g\in C_b((L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)),{w}))$ it holds

$$\begin{aligned} g(\mathcal {S}_\tau \mathbf {U}_m)\rightarrow g(\mathcal {S}_\tau \mathbf {U}),\qquad g(\mathbf {U}_m)\rightarrow g(\mathbf {U}). \end{aligned}$$

In particular, since every weakly continuous function is strongly continuous hence (A.2) holds with f replaced by g, we deduce by the dominated convergence theorem that

$$\begin{aligned} \mathbb {E}[ g(\mathcal {S}_\tau \mathbf {U})]=\mathbb {E}[ g(\mathbf {U})]. \end{aligned}$$

Now, it remains to verify the corresponding expression for a general strongly continuous function $f\in C_b(L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)))$. To this end, let $(\varphi _\varepsilon )$ be a smooth approximation to the identity on $\mathbb {R}\times \mathbb {T}^3$. Since convolution with $\varphi _\varepsilon$ is a compact operator on $L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$, we obtain that

$$\begin{aligned} \mathbf {U}\mapsto f(\mathbf {U}*\varphi _\varepsilon )=:f(\mathbf {U}^\varepsilon )\in C_b((L^q_{\mathrm{loc}}([0,\infty );W^{k,p}(\mathbb {T}^3)),{w})) \end{aligned}$$

and consequently

$$\begin{aligned} \mathbb {E}[ f(\mathbf {U}^\varepsilon )]=\mathbb {E}[ f([\mathcal {S}_\tau \mathbf {U}]*\varphi _\varepsilon )]=\mathbb {E}[f(\mathcal {S}_\tau \mathbf {U}^\varepsilon )], \end{aligned}$$

hence $\mathbf {U}^\varepsilon$ is stationary. Since

$$\begin{aligned} \mathbf {U}^\varepsilon \rightarrow \mathbf {U}\quad \text {in}\quad L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))\quad {\mathbb {P}}\text {-a.s.,} \end{aligned}$$

we may pass to the limit $\varepsilon \rightarrow 0$ and conclude using the dominated convergence theorem. $\square$

Lemma A.5

Let $k\in \mathbb {N}_0$, $p\in [1,\infty )$ and let $(\mathbf {U}_m)$ be a sequence of $W^{k,p}(\mathbb {T}^3)$-valued stochastic processes which are stationary on $W^{k,p}(\mathbb {T}^3)$ in the sense of Definition 2.7. If for all $T>0$

$$\begin{aligned} \sup _{m\in \mathbb {N}} \mathbb {E} \left[ \sup _{t\in [0,T]}\Vert \mathbf {U}_m\Vert _{W^{k,p}(\mathbb {T}^3)} \right] <\infty \end{aligned}$$

(A.3)

and

$$\begin{aligned} \mathbf {U}_m\rightarrow \mathbf {U}\quad \text {in}\quad C_{\mathrm {loc}}([0,\infty );(W^{k,p}(\mathbb {T}^3),w))\quad {\mathbb {P}}\text {-a.s.,} \end{aligned}$$

then $\mathbf {U}$ is stationary on $W^{k,p}(\mathbb {T}^3)$.

Proof

The claim is a consequence of Corollary A.3 and Lemma A.4. Indeed, as a consequence of (A.3) we deduce that

$$\begin{aligned} \mathbb {E} \left[ \sup _{t\in [0,T]}\Vert \mathbf {U}_m\Vert _{W^{k,p}(\mathbb {T}^3)} \right] <\infty \end{aligned}$$

thus $\mathbf {U}_m$ satisfies the assumptions of Corollary A.3 and the same is true for $\mathbf {U}$ due to lower semicontinuity of the corresponding norm. Accordingly, $\mathbf {U}_m$ satisfy the assumptions of Lemma A.4 which implies that $\mathbf {U}$ is stationary in the sense of Definition 2.8. Corollary A.3 then yields the claim. $\square$

Let us conclude with a simple observation that stationarity is preserved under composition with measurable functions.

Corollary A.6

Let $k\in \mathbb {N}_0$, $p\in [1,\infty )$. Let the stochastic process $\mathbf {U}$ be stationary on $W^{k,p}(\mathbb {T}^3)$ in the sense of Definition 2.7. Then for every measurable function $F:W^{k,p}(\mathbb {T}^3)\rightarrow \mathbb {R}$, the stochastic process $F(\mathbf {U})$ is stationary on $\mathbb {R}$.

Proof

The proof follows immediately from the corresponding equality of joint laws of $(\mathbf {U}(t_1),\dots , \mathbf {U}(t_n))$ and $(\mathbf {U}(t_1+\tau ),\dots , \mathbf {U}(t_n+\tau ))$. $\square$

Corollary A.7

Let $k\in \mathbb {N}_0$, $p,q\in [1,\infty )$. Let $\mathbf {U}$ be stationary on $L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ in the sense of Definition 2.8. Then for every measurable function $F:W^{k,p}(\mathbb {T}^3)\rightarrow \mathbb {R}$ and a.e. $s,t\in [0,\infty )$, the laws of $\mathbf {U}(s)$ and $\mathbf {U}(t)$ on $W^{k,p}(\mathbb {T}^3)$ coincide.

Proof

Since the mapping $\mathbf {U}\mapsto \mathbf {U}(t) \mapsto F(\mathbf {U}(t))$ is measurable on $L^q_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ for a.e. $t\in [0,\infty )$. For the same reasons, the mapping $\mathcal {S}_{s-t}:\mathbf {U}\mapsto \mathbf {U}(s) \mapsto F(\mathbf {U}(s))$ is measurable on $L^q_{\text {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ for a.e. $s,t\in [0,\infty )$. Hence the claim follows from the equality of laws of $\mathbf {U}$ and $\mathcal {S}_{s-t}\mathbf {U}$. $\square$

Remark A.8

Note that in view of Corollary A.7 the stationarity in the sense of Definition 2.8 implies the following almost everywhere version of Definition 2.7: if $\mathbf {U}$ is stationary on $L^q_{\mathrm {loc}}([0,\infty );W^{k,p}(\mathbb {T}^3))$ in the sense of Definition 2.8 then the joint laws

$$\begin{aligned} \mathcal {L}(\mathbf {U}(t_1+\tau ),\dots , \mathbf {U}(t_n+\tau )),\quad \mathcal {L}(\mathbf {U}(t_1),\dots , \mathbf {U}(t_n)) \end{aligned}$$

on $[W^{k,p}(\mathbb {T}^3)]^n$ coincide for a.e. $\tau \ge 0$, for a.e. $t_1,\dots ,t_n\in [0,\infty )$.