Abstract
In this chapter, we first show that the stochastic approximating manifold of a critical manifold as obtained in Volume I [41, Theorem 6.1] can be interpreted as the pullback limit associated with an auxiliary backward-forward system; see Eq. (4.1a–4.1c) and Lemma 4.1. The key idea consists of representing the modes with high wavenumbers as a pullback limit depending on the time history of the modes with low wavenumbers. We introduce then the concept of stochastic parameterizing manifolds (PMs), which are stochastic manifolds that improve in a mean square sense the partial knowledge of the full SPDE solution \(u\) when compared to the projection of \(u\) onto the resolved modes, for a given realization of the noise; see Definition 4.1. Backward-forward systems are also designed in Sect. 4.3 to give access to such stochastic PMs in practice. The resulting manifolds obtained by such a procedure are not subject to a spectral gap condition such as encountered in the classical theory of stochastic invariant manifolds. Instead, certain stochastic PMs can be determined under weaker non-resonance conditions. Such parameterizing manifolds will turn out to be useful in the design of efficient reduced models for the amplitudes of the resolved (low) modes of an SPDE solution, even when these amplitudes are large; see Chaps. 5, 6 and 7.
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Notes
- 1.
Such a manifold in our terminology is aimed to approximate some targeted stochastic invariant manifold (that can be local). It should not be confused with the notion of stochastic approximate inertial manifold [54] which makes sense even when no stochastic invariant manifold is guaranteed to exist.
- 2.
See also Corollary 4.1 for more precise rigorous results in the deterministic context.
- 3.
Using standard Landau notations. Note that there is a typo in the error bound reported in [23, Theorem. 7]. The correct error bound, when using their notations, should read \(\Vert h(\omega , \xi ) - e^{z(0)} L_s^{-1}B_s(\xi , \xi )\Vert \le C\big [R^2 + \big (\sigma K_2(\omega ) + R^2e^{2\sigma K(\omega )} + \sigma K_2(\omega )e^{2\sigma K(\omega )}\big )\Vert \xi \Vert + e^{2\sigma K(\omega )}\Vert \xi \Vert ^2\big ]\Vert \xi \Vert \), after combining the estimates derived by the authors in [23, Theorem. 6], Eqs. (39), (42), and (43) thereof.
- 4.
Not necessarily restricted to the case of stochastic unstable manifold considered in [45].
- 5.
- 6.
Since the system is autonomous, the two-time description of the dynamics \(S_{\xi }(t',s')\) reduces obviously to that given by a semigroup \(\tilde{S}_{\xi }(t' - s')\). We adopt however this way of writing the solution operator for the sake of unifying the different approximating manifolds discussed in this section.
- 7.
With global Lipschitz nonlinearities which do not cause a loss of regularity compared to the ambient space \(\fancyscript{H}\).
- 8.
Here various notions of random attractor could be used [61], for simplicity we can keep in mind the more standard one [60, 62].
- 9.
Typically, the \(m\) first eigenmodes of the linear part.
- 10.
Note that when a stochastic inertial manifold \(\varPhi \) exists for a given \(\fancyscript{H}^{\mathfrak {c}}\), then \(\varPhi \) provides obviously an optimal solution to this problem since for \(u=u_{\mathfrak {s}}+u_{\mathfrak {c}} \) on the global attractor, we have \(u_{\mathfrak {s}} (t,\omega )=\varPhi (u_{\mathfrak {c}}(t,\omega ),\theta _t \omega )\) in that case.
- 11.
The PES is no longer required here.
- 12.
With a slight abuse of language, the variance is understood here for a centered version of \(u_\mathfrak {s}\) with respect to its empirical average over \([0,T]\).
- 13.
which is equivalent to the existence of the integral in (4.45).
- 14.
associated with the SPDE (2.1) and the subspace \(\fancyscript{H}^{\mathfrak {c}}\).
- 15.
Note that, in the deterministic case, the corresponding backward-forward system (4.47a–4.47e) becomes autonomous and the equation for the \(\widehat{u}_{\mathfrak {s}}^{(n)}\) is solved over the interval \([-T, 0]\) instead of \([0, T]\), see for instance (4.19). Here, the definition of a parameterizing manifold for a deterministic PDE follows the same lines as Definition 4.1 with the \(\omega \)-dependence removed and the requirements regarding the measurability dropped.
- 16.
- 17.
Recall that the complexification of a real Hilbert space \(H\) is defined to be \(\widetilde{H}:=\{u+iv : u,v \in H\}\) endowed with the inner product \(\langle \widetilde{u}, \widetilde{v} \rangle _{\widetilde{H}}:= (\langle u_R, v_R \rangle _H + \langle u_I, v_I \rangle _H) + i (\langle u_I, v_R \rangle _H - \langle u_R, v_I \rangle _H)\), where \(\widetilde{u} = u_R + i u_I\) and \(\widetilde{v} = v_R + i v_I\) with \(u_R, u_I, v_R, v_I \in H\), and \(\langle \cdot , \cdot \rangle _H\) denotes the inner product on \(H\). The complexification of an operator \(G\) on \(H\) is defined to be \(\widetilde{G}:\widetilde{H} \rightarrow \widetilde{H}, u + i v \mapsto G (u) + i G(v)\).
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Chekroun, M.D., Liu, H., Wang, S. (2015). Pullback Characterization of Approximating, and Parameterizing Manifolds. In: Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-12520-6_4
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DOI: https://doi.org/10.1007/978-3-319-12520-6_4
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