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Exponential mixing for a class of dissipative PDEs with bounded degenerate noise

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Abstract

We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dual-Lipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a bounded random force which affects only a few Fourier modes. We assume that the nonlinear PDE in question is well posed, its nonlinearity is non-degenerate in the sense of the control theory, and the random force is a regular and bounded function of time which satisfies some decomposability and observability hypotheses. This class of forces includes random Haar series, where the coefficients for high Haar modes decay sufficiently fast. In particular, the result applies to the 2D Navier–Stokes system and the nonlinear complex Ginzburg–Landau equations. The proof of the abstract theorem uses the coupling method, enhanced by the Newton–Kantorovich–Kolmogorov fast convergence.

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Notes

  1. In the case of the Navier–Stokes system, the basis \(\{\varphi _i\}\) is composed of divergence-free \({{\mathbb {R}}}^2\)-valued functions on \({{\mathbb {T}}}^2\) with zero mean value, while in the case of the Ginzburg–Landau equation, they are complex-valued functions on \({{\mathbb {T}}}^3\).

  2. In the case when \(c_j=2^{j/2}\) for \(j\ge 1\) and \(\{\xi _k,\xi _{jl}\}\) are independent random variables with centred normal law of unit dispersion, the series (0.4) converges to the white noise; see Theorem 1 in [Lam96, Sect. 22]. Moreover, by Donsker’s invariance principle (see [Bil99, Sect. 8]), the integral of (0.4) converges to the Brownian motion on large time scales.

  3. The one-dimensional Burgers equation and multi-dimensional Burgers system in the potential setting can be treated by softer tools; see [Sin91, Bor16, Shi17].

  4. See the paper [KN13], where the mixing is proved for the white-forced cubic CGL equations, and the difficulty coming from nonlinearities of higher degree is explained.

  5. To see this, consider a measurable space \((Y,{{\mathcal {Y}}})\) and a measurable map \(A:Y\rightarrow {{\mathcal {L}}}(E,H)\) and denote by \({{\mathcal {G}}}_A\) the set of points \(y\in Y\) for which the image of A(y) is dense in H. Choosing countable dense subsets \(\{f_i\}\subset E\) and \(\{h_j\}\subset H\), it is easy to see that \({{\mathcal {G}}}_A=\bigl \{y\in Y:\inf _{i\ge 1}\Vert A(y)f_i-h_j\Vert _H=0 \text{ for } \text{ any } j\ge 1\bigr \}\). In the case under study, we have \(Y=H\times E\), \(A=(D_\eta S)(u,\eta )\), and \({{\mathcal {K}}}^u={{\mathcal {K}}}\cap {{\mathcal {G}}}^u\), where \({{\mathcal {G}}}^u=\{\eta \in E:(u,\eta )\in {{\mathcal {G}}}_A\}\).

  6. In a different form, this method as a tool to prove the exponential mixing for nonlinear PDEs with stochastic forcing was suggested in [KPS02]. In the final form (used in our work) the method was presented in [Kuk02].

  7. The first difficulty does not have an analogue in the KAM theory, whereas the second is usually present and manifests itself in the fact that the homological equation (1.19) cannot be solved for all actions \(p\in B\).

  8. In [KS12], the result is proved with \(\beta =1\). However, using (2.3), it is straightforward to check that the proof remains valid for any \(\beta \in (0,1)\).

  9. The role of inequalities (2.17) and (2.18) will be clarified below.

  10. We shall often omit the argument \(\omega \) to simplify formulas.

  11. Notice that we cannot apply the mean value theorem, since we do not know if \({{\mathcal {K}}}^{u,\sigma ,\theta }\) is convex.

  12. We emphasise that (\(\hbox {H}_4\)) can be regarded as a condition on the law \(\ell \) of the random variables \(\eta _k\), since any other random variable with law \(\ell \) has the same structure as \(\eta _k\).

  13. The concept of observability is widely used in the control theory and means, roughly speaking, that if a functional of a non-zero solution of a homogeneous linear differential equation vanishes identically in time, then it must be zero. In Definition 4.1, we have a similar property for functions: the left-hand side of (4.2) defines an affine function, and if it vanishes on \(\zeta \), then it must be zero.

  14. The convergence in \(L^1(J)\) of the series in (4.2) (when \({{\mathcal {I}}}\) is infinite) follows easily from (4.1).

  15. We emphasise that condition of saturation depends on the parameters \(a_1\) and \(a_2\) of the torus and the quadratic term Q, but not on the viscosity \(\nu \).

  16. The operator \(G^{{\tilde{u}}}\) is called the Gramian of the linear control problem associated with (4.6). It is a central object for studying controllability properties and enables one to formulate them in terms of observability of solutions for the dual problem; see Chapter 1 in [Cor07].

  17. Let us check briefly that the regularity of \({\tilde{u}}\) and w is sufficient to justify (4.11). Indeed, as was mentioned above, the function w belongs to \(W^{1,2}(J,H^{-1})\), so that the \(L^2\)-scalar product \((\zeta ,w(t))\) can be differentiated for any \(\zeta \in H^1\). Resolving the first equation in (4.8) with respect to \(\dot{w}\) and substituting the resulting expression into the time derivative of (4.10), we obtain \((\zeta ,\nu Lw+Q({\tilde{u}})^*w)=0\), where the equality is valid for almost every \(t\in J\). Since \({\tilde{u}}\in L^2(J,H^3)\) and \(w\in L^2(J,H^1)\), we can integrate by parts in the last equality, which gives the validity of equality (4.11) for almost every \(t\in J\). Finally, recalling that \({\tilde{u}}\in C(J,H^2)\) and \(w\in C(J,L^2)\), we see that the right-hand side is a continuous function of time and, hence, vanishes on J.

  18. A similar remark applies to the case of a rectangular domain with Lions boundary condition; see Sect. 9 in [AS08]. However, we shall not elaborate on that point since the analysis is similar.

  19. Note that the index j in (1.3) is now replaced by the pair (l, i), and vectors \(e_j\) are the products \(\psi _l^i(t)\varphi _i\).

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Acknowledgements

We thank V. I. Bogachev for discussion on the measurable version of gluing lemma, which was established in the paper [BM19a] on our request. We also thank the Institut Henri Poincaré in Paris for hosting our working group Control Theory and Stochastic Analysis, at which this article was initiated. This research was supported by the Agence Nationale de la Recherche through the Grants ANR-10-BLAN 0102 and ANR-17-CE40-0006-02. SK thanks the Russian Science Foundation for support through the Grant 18-11-00032. VN and AS were supported by the CNRS PICS Fluctuation theorems in stochastic systems. The research of AS was carried out within the MME-DII Center of Excellence (ANR-11-LABX-0023-01) and supported by Initiative d’excellence Paris-Seine.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS, Université Paris Diderot, UMR 7586, Sorbonne Paris Cité, 75013, Paris, France

    Sergei Kuksin

  2. School of Mathematics, Shandong University, Jinan, People’s Republic of China

    Sergei Kuksin

  3. Saint Petersburg State University, Universitetskaya nab., St. Petersburg, Russia

    Sergei Kuksin

  4. Laboratoire de Mathématiques, UMR CNRS 8100, UVSQ, Université Paris-Saclay, 45, av. des Etats-Unis, 78035, Versailles, France

    Vahagn Nersesyan

  5. Department of Mathematics, CY Cergy Paris Université, CNRS UMR 8088, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise, France

    Armen Shirikyan

  6. Centre de Recherches Mathématiques, CNRS UMI 3457, Université de Montréal, Montreal, QC, H3C 3J7, Canada

    Armen Shirikyan

  7. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    Armen Shirikyan

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  1. Sergei Kuksin
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Kuksin, S., Nersesyan, V. & Shirikyan, A. Exponential mixing for a class of dissipative PDEs with bounded degenerate noise. Geom. Funct. Anal. 30, 126–187 (2020). https://doi.org/10.1007/s00039-020-00525-5

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