Skip to main content

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.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Bensoussan, A.: Stochastic Navier–Stokes equations. Acta Appl. Math. 38(3), 267–304 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bourgain, J.: Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Commun. Math. Phys. 176, 421–445 (1996)

    MATH  Article  Google Scholar 

  3. 3.

    Breit, D., Feireisl, E., Hofmanová, M.: Incompressible limit for compressible fluids with stochastic forcing. Arch. Ration. Mech. Anal. 222, 895–926 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Breit, D., Feireisl, E., Hofmanová, M.: Compressible fluids driven by stochatic forcing: the relative energy inequality and applications. Commun. Math. Phys. 350, 443–473 (2017)

    MATH  Article  Google Scholar 

  5. 5.

    Breit, D., Feireisl, E., Hofmanová, M.: Local strong solutions to the stochastic compressible Navier–Stokes system. Comm. PDE. 43(2), 313–345 (2018)

  6. 6.

    Breit, D., Hofmanová, : Stochastic Navier–Stokes equations for compressible fluids. Indiana Univ. Math. J. 65, 1183–1250 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Brzeźniak, Z., Ferrario, B.: Stationary solutions for stochastic damped Navier–Stokes equations in \({\mathbb{R}}^d\). ArXiv e-prints (2017)

  8. 8.

    Brzeźniak, Z., Motyl, E., Ondreját, M.: Invariant measure for the stochastic Navier–Stokes equations in unbounded 2D domains. Ann. Probab. 45(5), 3145–3201 (2017)

  9. 9.

    Brzeźniak, Z., Ondreját, M.: Strong solutions to stochastic wave equations with values in Riemannian manifolds. J. Funct. Anal. 253, 449–481 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Burq, N., Tzvetkov, N.: Invariant measure for the three dimensional nonlinear wave equation. (2007). https://doi.org/10.1093/imrn/rnm108

  11. 11.

    Da Prato, G., Debussche, A.: Ergodicity for the 3D stochastic Navier–Stokes equation. J. Math. Pures Appl. 82, 877–947 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Da Prato, G., Debussche, A.: On the martingale problem associated with the 2D and 3D stochastic Navier–Stokes equation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei. 19, 247–264 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229, p. xii+339. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  14. 14.

    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

    MATH  Book  Google Scholar 

  15. 15.

    Debussche, A., Glatt-Holtz, N., Temam, R.: Local martingale and pathwise solutions for an abstract fluids model. Phys. D 14–15, 1123–1144 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Ebin, G.D.: Viscous fluids in a domain with frictionless boundary. Glob. Anal. Manifolds Teubner-Texte Math. 57, 93–110 (1983)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Feireisl, E.: Global attractors for the Navier–Stokes equations of three-dimensional compressible flow. C. R. Acad. Sci. Paris Ser. I 331, 35–39 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Feireisl, E., Maslowski, B., Novotný, A.: Compressible fluid flows driven by stochastic forcing. J. Differ. Equ. 254(3), 1342–1358 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid. Mech. 3, 358–392 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Feireisl, E., Petzeltová, H.: Bounded absorbing sets for the Navier–Stokes equations of compressible fluid. Commun. Partial Differ. Equ. 26, 1133–1144 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Feireisl, E., Pražák, D.: Asymptotic Behavior of Dynamical Systems in Fluid Mechanics. AIMS, Springfield (2010)

    MATH  Google Scholar 

  23. 23.

    Flandoli, F.: Dissipativity and invariant measures for stochastic Navier–Stokes equations. NoDEA 1, 403–423 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102(3), 367–391 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Flandoli, F., Romito, M.: Partial regularity for the stochastic Navier–Stokes equations. Trans. Am. Math. Soc. 354, 2207–2241 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Flandoli, F., Romito, M.: Markov selections for the 3D stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 140(3–4), 407–458 (2008)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Goldys, B., Maslowski, B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s. Ann. Probab. 34, 1451–1496 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Goldys, B., Maslowski, B.: Exponential ergodicity for stochastic Burgers and 2D Navier–Stokes equations. J. Funct. Anal. 226, 230–255 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, vol. 249, 2nd edn, p. xvi+489. Springer, New York (2008)

    MATH  Google Scholar 

  30. 30.

    Gyöngy, I., Krylov, N.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105(2), 143–158 (1996)

    MATH  Article  Google Scholar 

  31. 31.

    Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164(3), 993–1032 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Hieber, M., Prüss, J.: Heat kernels and maximal \(L^p-L^q\) estimates for parabolic evolution equations. Commun. Partial Differ. Equ. 22(9–10), 1647–1669 (1997)

    MATH  MathSciNet  Google Scholar 

  33. 33.

    Itô, K., Nisio, M.: On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ. 4, 1–75 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Jakubowski, A.: The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen 42(1), 209–216 (1997). (translation in Theory Probab. Appl. 42 (1997), no. 1, 167-174 (1998))

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Lanford, O.E.: The classical mechanics of one-dimensional systems of infinitely many particles. II. Kinetic theory. Commun. Math. Phys. 11, 257–292 (1968/1969)

  36. 36.

    Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford Science Publication, Oxford (1998)

    MATH  Google Scholar 

  37. 37.

    Romito, M.: Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system. Stochastics 82, 327–337 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Smith, S.: Random Perturbations of Viscous Compressible Fluids: Global Existence of Weak Solutions. SIAM J. Math. Anal., 49(6), 4521–4578 (2015)

  39. 39.

    Tornatore, E., Yashima, H.F.: One-dimensional stochastic equations for a viscous barotropic gas. Ric. Mat. 46(2), 255–283 (1998). 1997

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Tornatore, E.: Global solution of bi-dimensional stochastic equation for a viscous gas. NoDEA Nonlinear Differ. Equ. Appl. 7(4), 343–360 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Wang, D., Wang, H.: Global existence of martingale solutions to the three-dimensional stochastic compressible Navier–Stokes equations. Differ. Integr. Equ. 28(11–12), 1105–1154 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Martina Hofmanová.

Additional information

The research of Eduard Feireisl leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078. Bohdan Maslowski has been supported by GACR Grant No. 15-08819S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

Appendix A: Auxiliary results

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. \(t_1,\dots ,t_n\in [0,\infty )\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Breit, D., Feireisl, E., Hofmanová, M. et al. Stationary solutions to the compressible Navier–Stokes system driven by stochastic forces. Probab. Theory Relat. Fields 174, 981–1032 (2019). https://doi.org/10.1007/s00440-018-0875-4

Download citation

Keywords

  • Navier–Stokes system
  • Compressible fluid
  • Stochastic perturbation
  • Stationary solution

Mathematics Subject Classification

  • 60H15
  • 60H30
  • 35Q30
  • 76M35
  • 76N10