Abstract
In this paper, we construct invariant measures and global-in-time solutions for a fractional Schrödinger equation with a Moser–Trudinger type nonlinearity
on a compact Riemannian manifold M without boundary of dimension \(d\ge 2\). To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy (’19) and Sy and Yu (’21). More precisely, we show that if \(s>d/2\) or if \(s\le d/2\) and \(s\le 1+\alpha \), there exists an invariant measure \(\mu ^{s}\) and a set \(\Sigma ^s \subset H^s\) containing arbitrarily large data such that \(\mu ^{s}(\Sigma ^s ) =1\) and that (E) is globally well-posed on \(\Sigma ^{s}\). In the case when \(s>d/2\), we also obtain a logarithmic upper bound on the growth of the \(H^r\)-norm of our solutions for \(r<s\). This gives new examples of invariant measures supported in highly regular spaces in comparison with the Gibbs measure constructed by Robert (’21) for the same equation.
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1 Introduction
In this paper, we are interested in the construction of invariant measures and global-in-time solutions for the following fractional Schrödinger equation with a Moser–Trudinger type nonlinearity
Here \(\alpha \in (0,1]\) and \(\beta >0\) are fixed parameters, and M is a compact Riemannian manifold without boundary of dimension \(d\ge 1\).
This equation has been introduced in [16] for \(\alpha =1\) to describe a self-focusing laser beam whose radius is much greater than the vacuum wavelength. Equation (1.1) is also interesting from a purely mathematical point of view since when \(d=2\) the nonlinearity is energy critical, where the energy is
To illustrate this point, let us consider the fractional Schrödinger equation with pure power nonlinearity
Recall that it satisfies two conservation laws, in the sense that the mass and the p-energy
are preserved along the evolution, at least formally. Moreover, (1.3) enjoys a scaling invariance
and therefore the critical \(\dot{H}^{s_c}\) Sobolev regularity is given by the critical exponent
The power-NLS is said super-critical (resp. critical, resp. subcritical) if \(s_c >\alpha \) (resp. \(s_c = \alpha \), resp. \(s_c <\alpha \)). Viewing formally the exponential nonlinearity as an infinite-degree polynomial \(e^{\beta |u|^2}u= \sum \dfrac{\beta ^k}{k!} |u|^{2k} u\) we see that the critical Sobolev scale for (1.1) is given by \(p=\infty \), namely
As a consequence (1.1) is energy critical in dimension \(d=2\) for the standard diffusion \(\alpha =1\). When \(M=\mathbb {R}^2\), this equation has been intensively studied see [8, 12]. We also refer to [5, 19, 27] for related equations with the same type of nonlinearity. In particular, the global well-posedness of (1.1) for small initial data and defocusing nonlinearity has been obtained in [8] when \(M=\mathbb {R}^2\) and \(\alpha =1\). In the same setting, a scattering result for small data when \(\beta =4\pi \) was proved in [12]. On the other hand, when M is a compact manifold, it is interesting to investigate the behavior of (1.1) since scattering should not be expected. In fact, one expects that generic solutions should exhibit some energy cascade from low to high frequencies, which should presumably be detected by the unboundedness \(\limsup _{|t|\rightarrow \infty } \Vert u(t)\Vert _{H^s}=+\infty \) for large enough s. Our main goal in this paper is to construct some invariant measures, prove probabilistic global well-posedness for (1.1), and obtain some information about the long-time behavior of solutions (or rather, the possible growth rate of their Sobolev norms).
The first result in this direction was obtained by Bourgain [3] for a nonlinear Schrödinger equation in 1-d. The main obstruction to obtaining global solutions at this low level of regularity is the lack of conservation laws. In order to tackle this issue, Bourgain was able to use the Gibbs measure constructed by Lebowitz et al. [17] (supported on \(H^{1/2-}\)) in order to obtain bounds for some N-dimensional Galerkin projections of the equation (which were already used in the construction of the Gibbs measure). Most notably, these bounds were independent of the dimension N. Comparing the full problem with its finite-dimensional projections, Bourgain then managed to prove the global existence of solutions. To some extent, invariant measures allow conservation laws to survive even in low regularity, at least in a statistical sense.
Concerning our specific equation, Gibbs measures were constructed by [20]. More precisely, let us set up some notations. We denote
and
where \(\hat{u}_n =\langle u,e_n\rangle \) is the n-th Fourier coefficient in the \(L^2\) expansion of u on the Hilbert basis \(\{e_n\}_{n\in \mathbb {N}}\) given by the Laplace-Beltrami eigenfunctions with eigenvalues \(-\lambda _n\). Notice that \(\mu _\alpha \) is the law of the \(L^2\)-valued random variable
where \(g_n\) is a family of independent standard complex-valued Gaussians on some probability space \((\Omega ,\mathbb {P})\). In [20], it is shown that, if \(\alpha >d/2\) and \(s< \dfrac{2\alpha -d}{2}\),
is a well-defined probability measure supported on \(H^s (M)\) which is absolutely continuous with respect to \(\mu _\alpha \). Let us point out that the condition \(s< (2\alpha -d)/2\) is also necessary for the existence of the Gibbs measure. Thus the support \({{\,\textrm{supp}\,}}\rho _{\alpha ,\beta }\) necessarily corresponds to rough regularity. We will see in the following that it is possible to construct, by different means, some invariant measures supported in \(H^s\) for high regularities \(s>d/2\) or for \(s\le \min \{d/2, 1+\alpha \}\). However, these will not be defined explicitly as above.
Robert then uses this Gibbs measure in order to construct global solutions to (1.1). More precisely, let \(\alpha >d/2\): For small enough \(0<\beta <\beta _0\) and \(0<s<\alpha -\frac{d}{2}\), there exists a random variable u in \(C(\mathbb {R}; H^s)\) which solves (1.1) in the distributional sense, and \(\rho _{\alpha ,\beta }\) remains invariant over time. In the high regularity regime, one can in fact retrieve more information: If \(\alpha >d\), then for any \(0<\beta \ll \beta _0\) the evolution (1.1) is \(\rho _{\alpha ,\beta }\)-almost surely globally well-posed and the Gibbs measure \(\rho _{\alpha ,\beta }\) is flow-invariant. More precisely, for \(\frac{d}{2}<s<\frac{2\alpha -d}{2}\) and any \(\beta >0\), the flow is always locally well-posed, and there exists a set \(\Sigma \subset H^s\) of full \(\rho _{\alpha ,\beta }\)-measure such that, for any \(u_0 \in \Sigma \), the flow \(\phi ^t (u_0)\) is globally defined and \(\rho _{\alpha ,\beta }\) is invariant under \(\phi ^t\). Invariance is understood here in the sense that, for any \(\rho _{\alpha ,\beta }\)-measurable set \(A\subset \Sigma \), there holds \(\rho _{\alpha ,\beta }(\phi ^t(A))=\rho _{\alpha ,\beta }(A)\) for any \(t\in \mathbb {R}\). For low regularities \(d<\alpha \le 2d\), let us stress that even the local well-posedness is problematic (in particular uniqueness poses serious issues).
An alternative approach to the construction of Gibbs measures was developed by Kuksin [15] for the 2-d Euler equation, and adapted by Kuksin and Shirikyan [13] for the cubic Schrödinger equation. This is known as the fluctuation-dissipation method, and consists in considering a stochastic perturbation (converging to the original equation in the vanishing viscosity limit) having a stationary measure for any given viscosity parameter. Using a compactness argument, one wishes to show that this family of invariant measures has a weak limit as the viscosity goes to zero and that this limit is as expected an invariant measure for the original equation. A byproduct of the construction is suitable estimates (uniform in the level of noise) that can be used next to prove long-time existence results. We refer to [10, 14, 24,25,26] for related results.
Sy [22] and Sy and Yu [23] combined these two approaches into what they call Inviscid-Infinite-dimensional limits, and constructed invariant measures for a class of energy-supercritical NLS equations with pure-power nonlinearities. This method consists in using a fluctuation-dissipation argument on N-dimensional Galerkin approximations. In doing so, one obtains first a stationary measure for any fixed viscosity \(\sigma >0\) and finite dimension \(N\in \mathbb {N}\). An invariant measure for the original equation is then retrieved by taking first an inviscid limit \(\sigma \rightarrow 0\) and then an infinite-dimensional limit \(N\rightarrow \infty \). This is exactly the method that we implement in this work, with several technical improvements and difficulties arising here due to the exponential nonlinearity (instead of pure powers).
For our results we focus on the energy critical and supercritical cases only \(\alpha \le d/2\). In the high regularity regime \(s>d/2\) our main result reads
Theorem 1.1
Let \(d\ge 2\), \(\alpha \le d/2\), and fix \(s>d/2\). There exist an increasing concave function \(\zeta :\mathbb {R}^+ \rightarrow \mathbb {R}^+\), a probability measure \(\mu ^{s}\in \mathfrak {p}(L^2)\) supported on \(H^s\) and a set \(\Sigma ^{s} \subset H^{s}\) such that
-
\(\mu ^{s} (\Sigma ^{s} )=1\)
-
Equation (1.1) is globally well-posed on \(\Sigma ^{s}\)
-
The induced flow \(\phi ^t\) leaves the measure \(\mu ^{s}\) invariant
-
We have
$$\begin{aligned} \int _{L^2} \Vert u\Vert _{H^s}^2 \mu ^{s} (du) <\infty . \end{aligned}$$ -
The set \(\Sigma ^{s}\) contains data of arbitrarily large size, \(\mu ^{s} (\{\Vert u\Vert _{H^s}>K \})>0\) for all \(K>0\)
-
For any fixed \(r<s\) and all \(u_0\in \Sigma ^s\) we have
$$\begin{aligned} \Vert \phi _t (u_0)\Vert _{H^{r}} \le C \zeta (1+\ln (1+|t|)) \qquad \text {for all } t\in \mathbb {R}, \end{aligned}$$for some \(C=C(u_0,r)\).
Upon closer inspection, our construction actually yields a rough but explicit upper bound of the form \(\zeta (z)\sim C\sqrt{ |z|}\) for large |z| and some C depending on the various parameters. (See the proof of Proposition 3.8 for details.)
In the low regularity regime \(s\le d/2\) we will establish
Theorem 1.2
Let \(d\ge 2\), \(\alpha \le 1\), and fix \(s\le \min \{d/2, 1+\alpha \}\). There exists a probability measure \(\mu ^{s}\in \mathfrak {p}(L^2)\) supported on \(H^s\), a probability space \((\Omega ,\mathcal {F},\mathbb {P})\), and a \(L^2_{loc}(\mathbb {R};H^s)\cap C(\mathbb {R};H^{s-\alpha })\)-valued stochastic process u such that
-
\(u=u^\omega \) is a global distributional solution of (1.1) for \(\mathbb {P}\)-a.a. \(\omega \in \Omega \);
-
\(\mu ^{s}\) is invariant in the sense that the law of u(t) is \(\mu ^{s}\) for all \(t\in \mathbb {R}\)
-
We have
$$\begin{aligned} \int _{L^2} \Vert v\Vert _{H^s}^2 \mu ^{s} (dv) <\infty . \end{aligned}$$ -
\({{\,\textrm{supp}\,}}\,\mu ^{s}\) contains data of arbitrarily large size, \(\mu ^{s} (\{\Vert u\Vert _{H^s}>K \})>0\) for all \(K>0\).
Remark 1.3
The \(C(\mathbb {R};H^{r-\alpha })\) continuity of a.a. trajectory \(u^\omega \) appears here for technical convenience, and it seems plausible that trajectories should in fact be continuous for the \(H^s\) topology.
In this work we only address the existence of such invariant measures and the associated almost-sure global well-posedness. An interesting and more difficult question is to derive qualitative properties of these measures, e.g. the dimensionality of the supports and absolute continuity with respect to the Gaussian measure (1.4), and we leave these issues for future work.
Although the statement of Theorem 1.2 is very similar in spirit with Theorem 1.1, it is worth pointing out that the very notion of invariance is slightly different and that the rigorous statement is more probabilistic in nature than for high regularity. Indeed, for \(s>d/2\) the global solutions in Theorem 1.1 will be somehow constructed by hand, and the induced flow \(u(t)=\phi ^t (u_0)\) is at least deterministically well-posed. In particular, the flow is unique and is continuous with respect to initial data (see Lemma 2.4). However, at the very low level of \(H^s\) regularity involved in Theorem 1.2 above, even uniqueness of solutions to (1.1) may fail (in addition to existence), so the very notion of local flow is delicate. The stochastic setting here somehow allows for an implicit selection of solutions (arising from a passage to the infinite-dimensional limit \(N\rightarrow \infty \) in our construction), hence the statement does not even mention flow-maps and must be weakened in order to tackle this issue. Apart from that, the only but major difference compared with Theorem 1.1 is that we lose quantitative control over the \(H^r\) growth of solutions. An interesting question would be to investigate the uniqueness and the continuity of the flow in the limit case \(s=\frac{d}{2}\). These properties have been proved to hold in [21] when \(\alpha =1\) and \(M=\mathbb {R}^2\) (so that \(s=1\)). An important ingredient of the proof is the use of Strichartz estimates. However it is well-known that they a priori only hold with a regularity loss on compact manifolds [2]. So the proof of [21] does not seem to apply anymore.
Let us also observe that Theorem 1.1 holds in the whole supercritical range \(\alpha \in (0,d/2]\). This is no longer the case for Theorem 1.2, since the proof relies at some point on a Córdoba-Córdoba type lemma crucially requiring \(\alpha \le 1\). This restriction may seem surprising at first sight, as one expects that higher dispersion should simplify the analysis, and indeed this is an artificial obstacle due to our specific construction. We point out that, also for purely technical reasons, we were not able to cover the full range of Sobolev exponents in the low regularity regime, and \(s\in (1+\alpha ,d/2]\) is not covered in Theorem 1.2 if \(d/2>1+\alpha \). The very same artificial gap also appears in Sy’s work [24] for the same reasons.
As far as optimality is concerned, let us stress that (i) for the unregularized equation (1.1) there are a priori only two conserved quantities E(u), M(u), (ii) given the choice of a dissipation operator \(\mathcal {L}=\mathcal {L}_s\) involved in our construction by fluctuation-dissipation (to be described shortly), the best one can hope to control is the variation \(\mathcal {E},\mathcal {M}\) of E, M in the direction of \(\mathcal {L}_s\), and (iii) our current choices of \(\mathcal {L}_s\) control at best the \(H^s\) norm, and no better. This strongly suggests that our measures constructed in Theorems 1.1 and 1.2 should be somehow optimal, in the sense that they should satisfy \(\int _{L^2} \Vert u\Vert _{H^r}^2 \mu ^{s} (du)=\infty \) for any \(r>s\) and should thus be \(H^s\) critical for given s. (We will not discuss any further this delicate optimality issue in this work.)
1.1 Structure of the proof
Let us now describe informally our Inviscid-Infinite-dimensional construction, based on Sy [22] and Sy and Yu [23]. Let \(E_N\) be the space generated by the first N eigenvalues of the Laplace-Beltrami operator on M, and \(P_N\) the orthogonal projector thereupon. On \(E_N\), we define a Brownian motion and white noise by
where \((a_m)_{m\in \mathbb {Z}}\) is an arbitrary family of complex numbers (to be well-chosen later on and satisfying some decay conditions) and \(\beta _m\) is a sequence of independent standard real Brownian motions. We consider the following SPDE
where \(\sigma >0\) is a small viscosity parameter. Here \(\mathcal {L}\) is a (deterministic) dissipation operator to be well-chosen, and the Brownian motion of course induces stochastic fluctuations in the system. Clearly as \(\sigma \rightarrow 0\) and \(N\rightarrow \infty \) this is a perturbation of (1.1), and note that scaling \(\sigma ^2\mathcal {L}\) and \(\sigma \eta _N\) is consistent with Itô’s formula. For technical reasons we will choose different dissipation operators \(\mathcal {L}\), tailored to either strong or low regularity regimes. In any case the choice of \(\mathcal {L}=\mathcal {L}_s\) will be subordinated to the prior choice of a fixed Sobolev exponent \(s>0\), thus targeting any arbitrary \(H^s\) regularity.
We will prove that (1.5) is always stochastically well-posed on \(E_N\) (see Definition 3.1), and standard finite-dimensional Bogoliubov-Krylov arguments will allow to produce a stationary measure \(\mu ^s_{\sigma ,N}\) on \(E_N\). The energy and mass dissipation
in the direction of \(\mathcal {L}\) will be our two fundamental functionals, yielding barely sufficient compactness and \(H^s\) control throughout the whole analysis. More precisely, Itô’s formula will yield
uniformly in \(\sigma ,N\). Owing to our specific choices of \(\mathcal {L}\) for strong or low regularities, the resulting functionals \(\mathcal {E},\mathcal {M}\) will enjoy suitable coercivity and give just enough compactness on the family of stationary measures \(\left\{ \mu ^s_{\sigma ,N}\right\} _{\sigma ,N}\). Taking first the limit \(\sigma \rightarrow 0\) and then \(N\rightarrow \infty \), we establish the weak convergence of \(\mu ^s_{\sigma ,N}\) (in \(H^r\) for all \(r<s\)) to a measure \(\mu ^s\) satisfying the same estimates. Fine-tuning the Brownian coefficients \((a_m)_m\), one can moreover slightly modify \(\mu ^s\) in order to get initial data with arbitrarily large \(H^s\) norm, which guarantees in particular that \(\mu ^s\) is not trivially concentrated on the zero solution.
At this stage it remains to construct global solutions for a large-enough class of initial data \(u_0\in \Sigma ^s\) (with \(\mu ^s(\Sigma ^s)=1\)), and this is when the construction starts to really differ for strong and low regularities.
For high regularities \(s>d/2\) we choose to include a strong damping term \(\mathcal {L}(u)=(\dots )+G(\Vert u\Vert _{H^s})u\) in the dissipation operator, for some function G growing sufficiently fast in order to overtake the exponential nonlinearity. The Sobolev embedding \(H^s\subset L^\infty \) then allows to control the \(H^s\) norms of the projected solutions uniformly in \(N,\sigma \). This puts us in position of implementing Bourgain’s strategy [3] to construct a full \(\mu ^s\)-measure set \(\Sigma ^s \subset H^s\) on which (1.1) is globally well-posed. The invariance of \(\mu ^s\) will naturally follow from some strong convergence of the projected flow-map \(\phi ^N_t\) towards its unprojected counterpart \(\phi ^t\). Let us point out that this construction also provides a quantitative estimate on the slow logarithmic growth of the \(H^r\)-norm of the global solutions constructed here, inherited from the control for finite N.
For low regularities \(s\le d/2\) the failure of the Sobolev embedding \(H^s\not \subset L^\infty \) prevents from controlling the nonlinearity in \(H^s\), thus one cannot expect to control the growth of solutions even for the projected equation. In order to circumvent this issue we choose now to include the nonlinearity \(\mathcal {L}(u)=(\dots ) + P_N\left( u e^{\beta |u|^2}\right) \) in the dissipation operator. This will result in coercivity \(\mathcal {M}(u)\gtrsim \int |u|^2e^{\beta |u|^2}\) in the mass dissipation functional, which in turn will allow a statistical control \(\int _{L^2} \left\| u e^{\beta |u|^2}\right\| _{L^1_{t,x}}\mu ^{s}_{\sigma ,N}(du)\) for the nonlinearity in any finite time interval. However, some residual crossed terms can only be controlled via a suitable Cordóba-Cordóba inequality requiring \(s\le \alpha +1\), and our analysis unfortunately does not cover the range \(s\in (\alpha +1,d/2]\) (we were simply not able to find a dissipator \(\mathcal {L}\) allowing to control the nonlinearity while dominating the crossed terms.) At this stage, a soft but very flexible Skorokhod-type argument by Burq et al. [4] allows to conclude that \(\sigma ^s=\lim \sigma ^s_{\sigma ,N}\) is invariant, as the limit of invariant measures for which the nonlinearities \(P_N(u_Ne^{\beta |u_N|^2})\rightarrow ue^{\beta |u|^2}\) enjoy sufficient convergence. We stress that, in this case, we do not obtain quantitative control on the growth of \(H^r\)-norm of the solutions. As already pointed out, this is the only significant difference between Theorems 1.2 and 1.1.
The plan of this paper is the following: in Sect. 2, we establish several basic properties satisfied by solutions of finite-dimensional Galerkin projections of (1.1). In particular, we prove existence of a unique global solution in \(C(\mathbb {R};H^s)\). Moreover, for high regularities \(s>d/2\) we derive suitable quantitative bounds on the \(H^s\) norms in fixed time interval, uniformly in N. In the same regime, we also obtain some continuity properties of the flow-map \(\phi ^t\) associated with (1.1) and of its projected approximation.
Section 3 focuses on high regularities \(s>d/2\) and actually contains the core of the paper: we carry out the full technical details for the above construction and prove Theorem 1.1. This section is the most involved of the paper, as most of its material will be recycled in the next section (mostly, the construction of the invariant measure by compactness arguments).
Section 4 is devoted to low regularities and Theorem 1.2. As most of the technical details are similar to the previous section, we will not give the full details for the construction of the measure \(\mu ^s\). We will of course highlight the differences and give a full proof for the significantly different construction of global solutions as well as for the invariance of the measure.
Finally, some technical and auxiliary results are collected in the appendix.
2 Preliminaries and Galerkin projections
Let \(\lambda _0 \le \lambda _1 \le \ldots \le \lambda _m \le \ldots \) be the eigenvalues of the Laplace-Beltrami operator \(-\Delta \) on M. For notational convenience we write \(\lambda _{-m}=\lambda _m\). We denote by \((e_m)_{m\in \mathbb {N}}\) the corresponding eigenfunctions and by \(e_{-m}\) the eigenfunction \(i e_m\), so that \((e_m)_{m\in \mathbb {Z}}\) is a Hilbert basis for \(L^2 (M ;\mathbb {C})\).
We denote by \(\Vert .\Vert \) the standard \(L^2\) norm with corresponding scalar product
where \(\mathcal {R}\) stands for the real part. For \(u=\sum u_m e_m\) expressed in this Hilbert basis, the standard Sobolev norm reads
Let now \(E_N\) be the subspace of \(L^2\) spanned by \((e_m)_{|m|\le N}\), and let \(P_N\) be the orthogonal projection onto \(E_N\). On the finite-dimensional space \(E_N\), we have the norm equivalence
and
for any \(q\in (2,\infty ]\) and \(s\in (0,\infty )\).
For \(\gamma >0\) the spectral powers of the Laplacian are defined as
and we recall that \(P_N\) and \((-\Delta )^\gamma \) commute (by definition).
On \(L^2\) we define the dispersive propagator S(t) by
an we recall that it is unitary in the sense that
In this setting, we consider now the Galerkin projection of (1.1)
which is nothing but a nonlinear ODE on the finite-dimensional space \(E_N\) (note that \((-\Delta )^{\alpha }\) maps \(E_N\) to itself).
Let us start with various technical preliminaries on the well-posedness of (1.1) and (2.1). For fixed \(T>0\) and \(r\ge 0 \) we write
Proposition 2.1
Fix \(s>d/2\). For any \(R>0\), there exists a time \(T>0\) such that, for every \(N\in \mathbb {N}\) and \(u_0 \in B_R (H^s)\), there exists a unique solution \(u \in X_T^s\) to (2.1). This time can be taken of the form
for some small c and large C depending on s only. Moreover, this solution is in fact global-in-time and satisfies
The important point here is that we have a quantitative control (2.3) on the \(H^s\) norm in any fixed time interval uniformly in N.
Proof
Let \(u_0 \in B_R (H^s )\). For \(u\in X^s_T\) we set
and observe of course that u solves (2.1) with initial datum \(P_Nu_0\) if and only if it is a fixed point for \(F(u)=u\). We have, using the fact that S is unitary in \(H^s\) and Lemma A.1,
Taking \(T \lesssim \dfrac{c}{ \beta e^{C\beta R^2}}\) for c small enough and C large enough (only depending on s), we see that \(\Vert F(u)\Vert _{X^s_T}\le \Vert u_0\Vert _{H^s}+\frac{1}{2}\Vert u\Vert _{X^s_T}\le 2R\) and thus F maps \(B_{2R}(X^s_T)\) to itself.
Let now \(u_1,u_2\) be two functions in \(B_{2R} (X_T^s)\). By Lemma A.1 and because S(t) is unitary on \(H^s\), we have
This shows that F is a contraction on \(B_{2R}(X_T^s)\), again as soon as \(T \approx \dfrac{c}{e^{C\beta R^2}}\) for some small \(c>0\) and large \(C>0\) depending on s only. By the Banach fixed point Theorem we conclude that there exists a unique solution u to (2.1) in \(C([-T ,T] ; B_{2R}(H^s))\).
Finally, at least for \(u\in E_N\) there holds \(\left( i(-\Delta )^{\alpha } u,u\right) =0\) and \(\left( i P_N \left( u e^{\beta |u|^2}\right) ,u \right) =\left( i u e^{\beta |u|^2},u \right) =0\), whence the conservation of mass \(\dfrac{d}{dt}\Vert u\Vert ^2=2\left( u,\partial _tu\right) =0\) along solutions of (2.1). On the finite-dimensional space \(E_N\) this guarantees that u(t) remains in a fixed compact set (a ball of radius \(\Vert P_Nu_0\Vert \)), hence blowup in finite time is excluded and u(t) is thus global-in-time. \(\square \)
In the low regularity setting we have
Proposition 2.2
Assume that \(s\le d/2\). For any \(u_0 \in B_R (H^s)\) there exists a unique global solution \(u\in C (\mathbb {R}; H^s )\) to (2.1).
The main and important difference with Proposition 2.1 is that, for such low regularity, we lose quantitative \(H^s\) estimates in uniform time intervals such as (2.3). Some control can still be retrieved in intervals \([-T_N,T_N]\), but with \(T_N\) exponentially small as \(N\rightarrow \infty \). This is reminiscent from the fact that, for low regularity \(s<d/2\), the well-posedness of (1.1) for \(N=\infty \) is much more delicate.
Proof
The proof is quite similar to the previous one, with the major difference that \(H^s\) does not embed into \(L^\infty \) anymore. Instead, we use the norm equivalence \(\Vert u \Vert _{L^\infty } \le CN^{\frac{d}{2} -s} \Vert u\Vert _{H^s} \) for \(u\in E_N\) [20, Lemma 3.3] in order to get similarly
Taking \(T \lesssim \dfrac{c}{N^{d/2 -s} e^{CN^{d-2s}\beta R^2}}\) for suitably chosen constants \(c,C>0\), we have \(\Vert F(u)\Vert _{X_T^s} \le 2R\) for all \(u\in B_{2R} (X_T^s )\) and F thus maps \(B_{2R}(X_T^s)\) to itself. (Note that the upper bound for T is exponentially small as \(N\rightarrow \infty \).) The rest of the proof follows similarly. \(\square \)
Let us now turn to the unprojected Eq. (1.1) for \(N=\infty \). For \(s>d/2\) we have local well-posedness:
Proposition 2.3
Assume that \(s>d/2\). For any \(R>0\), there exists a time \(T>0\) such that, for every \(u_0 \in B_R (H^s)\), there exists a unique solution \(u\in X_T^s =C ([-T,T]; H^s )\) to (1.1) satisfying moreover
The time \(T=T(s,R)\) can be taken exactly as in (2.2)
The proof is identical to that of Proposition 2.1 (a fixed point argument heavily relying on Lemma A.1) and we omit the details. The important point is here that the local existence time \(T=T(s,R)\) is identical for \(N<\infty \) and \(N=\infty \). Of course, even in this strongly dispersive regime \(s>d/2\) we do not claim global existence at this stage (since ultimately this is the whole purpose of this paper, for as large a class of initial data as possible).
The rest of this section is devoted to further properties of the flows for \(s>d/2\). (For \(s\le d/2\) even the local well-posedness is unclear, in particular uniqueness is an issue.)
We denote by \(\phi ^t\) and \(\phi _N^t\) the corresponding flows associated with (1.1) and (2.1) (local and global-in-time, respectively), implicitly well-defined according to Propositions 2.1 and 2.3.
Lemma 2.4
Fix \(r>d/2\), let \(u_0\in B_R(H^r)\) for some \(R>0\), and choose \(T=T(r,2R)\) as in (2.2). If \(u_{0,N}\in E_N\) is any sequence such that \(u_{0,N}\rightarrow u_0\) in \(H^r\) then
Moreover both flows are locally Lipschitz continuous, in the sense that if \(u_0,u'_0\in B_R(H^r)\) and \(T=T(r,R)\) as before, then
and
for some \(C=C(R,r)\).
Proof
Let us begin with (2.5). Note first that, if N is large enough, then \(u_0,u_{0,N}\in B_{2R}\) hence \(\phi ^t(u_0)\) and \(\phi ^t_N(u_{0,N})\) are both defined at least for times \(|t|\le T=T(r,2R)\) and both remain in \(B_{4R}(H^r)\) (Propositions 2.1 and 2.3). Using Duhamel’s formula, we have, for any \(t\in [0,T)\)
Taking the difference, we get
Using that S(t) is unitary in \(H^r\), exploiting Lemma A.1 to control the last integral, and recalling that \(\phi ^\tau (u_0),\phi ^\tau _N(u_{0,N})\) remain contained in a fixed ball \(B_{4R}(H^r)\) for times \(\tau \in [-T,T]\), we get
Because \(f(\tau )=\phi ^\tau (u_0)e^{\beta |\phi ^\tau (u_0) |^2}\in H^r\) it is easy to see, for fixed \(u_0\), that we have pointwise convergence
as the remainder \(\sum _{|m|>N}(\dots )\) of an absolutely convergent series. The uniform bound \(\phi ^\tau (u_0)\in B_{4R}(H^r)\) combined with Lemma A.1 allows to conclude by Lebesgue’s dominated convergence that
Our previous estimate (2.8) thus reads
with \({\varepsilon }_N=\Vert u_0-u_{0,N}\Vert _{H^r} + I_N\rightarrow 0\) as \(N\rightarrow \infty \). Grönwall’s inequality leads to
and (2.5) follows.
For (2.6) we take the difference
Because we know that \(\phi ^\tau (u_0),\phi ^\tau (u_0')\in B_{2R}(H^r)\) for all \(|\tau |\le T\), and S(t) is unitary, our Lemma A.1 allows to control
for some \(C=C(R,r)\), and our claim immediately follows by Grönwall’s inequality.
The proof of (2.7) follows along the exact same lines and we omit the details. \(\square \)
For technical purposes we will also need to upgrade the convergence of \(\phi ^t_N\) towards \(\phi ^t\) to uniform convergence on compact sets
Corollary 2.5
Let \(r>d/2\) and fix any \(H^r\)-compact set K. There exists \(T=T(r,K)\) such that for any fixed \(|t|\le T\) we have uniform convergence
Proof
By compactness, we have first \(K\subset B_R(H^r)\) for some large enough \(R>0\). Let \(T=T(r,R)\) be as in Propositions 2.1 2.3, so that \(\phi ^t(u_0),\phi ^t_N(u_0)\) are well-defined and remain contained in \(B_{2R}(H^r)\) for all \(|t|\le T\). By the previous Lemma 2.4 we see that \(\phi ^t,\phi ^t_N:B_{R}(H^r)\rightarrow H^r\) are equi-Lipschitz, uniformly in N. Moreover, taking in particular \(u_{0,N}=P_Nu_0\) in (2.5), we see that \(\phi ^t_N(u_0)\rightarrow \phi ^t(u_0)\) pointwise in \(u_0\in B_R(H^r)\). A variant of the Arzelà -Ascoli theorem (Lemma A.8 in the Appendix) finally guarantees that the convergence is in fact uniform on K. \(\square \)
3 High regularity
In this section we focus exclusively on the high regularity regime \(s>d/2\), and recall that we only consider the energy critical and supercritical cases \(\alpha \le d/2\). The reader can think of the \(H^s\) scale as being fixed once and for all in this section, and we will rather use the \(H^r\) notation for any other Sobolev spaces. Let us first set-up our fluctuation-dissipation framework.
3.1 Fluctuation-dissipation and a priori estimates
On \(E_N\), we define a Brownian motion and a white noise by
where \((a_m)_{m\in \mathbb {Z}}\) is an arbitrary family of complex numbers (satisfying some decay conditions to be precised soon), \(\beta _m\) is a sequence of independent standard real Brownian motions with respect to a filtration \((\mathcal {F}_t )_{t\ge 0}\) and defined on a probability space \((\Omega , \mathcal {F} , \mathbb {P})\).
Since \(s>d/2\) and \(\alpha \in (0,d/2]\) is fixed, we have in particular \(s>\alpha \). We define the dissipation operator
for some smooth, nondecreasing function \(G:\mathbb {R}^+\rightarrow \mathbb {R}^+\)
for some small \(c>0\) and large \(\Lambda >0\). Here one can think of \(\Lambda \) to be as large as needed, in fact we will at times adjust \(\Lambda \ge C\) for various constants \(C=C(s)\) related to the Sobolev embedding via the exponential constant in Lemma A.1.
For some small parameter \(\sigma >0\) and fixed exponent \(\alpha >0\), consider the Stochastic PDE
We think of \(\mathcal {L}\) as a deterministic regularizing dissipation (hence the minus sign), and the white noise induces stochastic fluctuations in the system. As already pointed out, the scaling \(\sigma ^2\mathcal {L}\) and \(\sigma \eta _N\) is consistent with Itô’s formula and will be crucial in order to retrieve enough compactness estimates and take the limit \(\sigma \rightarrow 0^+\) later on.
By solutions of the SPDE, we mean here:
Definition 3.1
(Stochastic global well-posedness) We say that (3.4) is stochastically globally well-posed on \(E_N\) if the following properties hold:
-
(i) for any \(E_N\)-valued random variable \(u_0\) independent of \(\mathcal {F}_t\)
-
For \(\mathbb {P}\)-a.a. \(\omega \) there exists a solution \(u=u^\omega \in C(\mathbb {R}^+ ;E_N)\) of (3.4) with initial datum \(u_0=u_0^\omega \) in the integral sense, i-e
$$\begin{aligned} u(t)=u_0+\int _0^t \left[ -i\left( (-\Delta )^{\alpha }u + P_N\left( 2\beta u e^{\beta |u|^2}\right) \right) -\sigma ^2\mathcal {L}(u) \right] d\tau +\sigma \zeta _N(t) \end{aligned}$$(3.5)with equality as elements of \(C(\mathbb {R}^+;E_N)\).
-
If \(u_1^\omega ,u^\omega _2 \in C (\mathbb {R}^+; E_N )\) are two such solutions with the same initial data \(u^{\omega }_{1,0}=u^{\omega }_{2,0}\) then \(u_1^\omega \equiv u_2^\omega \).
-
-
(ii) (pathwise continuity w.r.t. initial data) for \(\mathbb {P}\)-a.a. \(\omega \in \Omega \), we require that
$$\begin{aligned} \lim _{u_0 \rightarrow u_0'} u^\omega (.\,; u_0) = u^\omega (.\,; u_0' )\ \quad \text{ in }\quad \ C_{\textrm{ loc}}(\mathbb {R}^+ ;E_N), \end{aligned}$$where \(u_0\) and \(u_0'\) are deterministic data in \(H^s\) and \(u^\omega (.\,; u_0),u^\omega (.\,; u_0')\) are the corresponding solutions of (3.5) for fixed \(\omega \)
-
(iii) The \(E_N\)-valued stochastic process \((\omega ,t) \mapsto u^\omega (t) \) is adapted to the filtration \(\sigma (u_0 , \mathcal {F}_t).\)
Owing to (i), any such solution, if any, is unique. Also, such a process solves the finite-dimensional forward Itô SDE
in the classical sense.
Proposition 3.2
Equation (3.4) is stochastically globally well-posed on \(E_N\) in the sense of Definition 3.1.
Proof
We will construct the solution in the form \(u=z+v\), where z will first contain all the Brownian fluctuation and v will then be obtained by solving pathwise a deterministic ODE with coefficients depending on z.
To this end let us first introduce
This is well-defined for \(\mathbb {P}\)-a.a. \(\omega \in \Omega \) as a function of time with values in \(E_N\), and it is the unique solution to the Itô SDE
We also have that \(z^\omega (.)\) belongs to \(C(\mathbb {R}^+ ;E_N)\) for \(\mathbb {P}\)-a.a. realization. Because \(\Vert z\Vert ^2\) is a submartingale (being z a martingale), Doob’s maximal inequality and Itô’s formula \(d\Vert z\Vert ^2=2\sigma (z,d\zeta _N)+\sigma ^2 dt\) (with \((i(-\Delta )^\alpha z ,z)dt=0\)) yield moreover, for any \(T>0\),
For any fixed \(\omega \in \Omega \) we consider now the deterministic ODE
where \(u_0=u_0^\omega \) and \(z=z^\omega (t)\) are considered as given, deterministic data for fixed \(\omega \). A straightforward application of the Cauchy-Lipschitz theorem guarantees existence and uniqueness of a local-in time solution \(v=v^\omega (t)\), and we will show below that the latter is in fact globally defined for \(t\in \mathbb {R}^+\). Also, observe that \(u=v+z\) will automatically solve our integral formulation (3.5) for fixed \(\omega \).
As a first step, let us prove that, \(\mathbb {P}\)-almost surely, the solution \(v=v^\omega \) of (3.9) is global-in-time. Using (3.9) as well the conservation laws \((v,i(-\Delta )^\alpha v)=0\) and \(\left( u, iP_N(ue^{\beta |u^2|})\right) =\left( u,iue^{\beta |u^2|}\right) =0\) for \(u=v+z\in E_N\), we get
Discarding the nonnegative term \(\Vert v\Vert ^2_{\dot{H}^{s-\alpha }}\) and using repeatedly Young’s inequality we control next
where we made a crucial use of our growth assumption (3.3) with \(\Lambda \ge C(s)\) in the last inequality.
Observe that, due to (3.8), for \(\mathbb {P}\)-almost all \(\omega \in \Omega \), there exists a constant \(C_{\omega ,T}\) such that \(z=z^\omega \) satisfies some bound
For fixed \(t\in [0,T]\) and \(\omega \in \Omega \), we have in(3.10) that either \(\Vert v(t)\Vert ^2\le c+Z_{\omega ,T}^2\), or \(\Vert v(t)\Vert ^2\ge c +Z^2_{\omega ,T} \ge c+\Vert z(t)\Vert ^2\). In this latter case, the last term in (3.10) is nonpositive and we find that \(\dfrac{d}{dt}\left( \frac{1}{2}\Vert v\Vert ^2\right) \le \dfrac{8\beta ^2Z^2_{\omega ,T}}{\sigma ^2} \le 8\beta ^2C_{\omega ,T} \). Owing to the rough bound \(Z_{\omega ,T}=\sigma C_{\omega ,T}\le C_{\omega ,T}\) for \(\sigma \le 1\), this readily gives an a-priori estimate
for a.a. \(\omega \in \Omega \), hence blowup in finite time is excluded and v is indeed almost surely global-in-time. Note that this pathwise estimate is independent of \(\sigma \), which will be important later on when we take \(\sigma \rightarrow 0\) (in particular for Proposition 3.6).
Next, let us prove uniqueness. Assume \(u_i=u_i^\omega \), \(i=1,2\) are two solutions of (3.5) with deterministic initial data \(u_{0,i}\). For \(z=z^\omega (t)\) as in (3.6)(3.7) and a.a. \(\omega \in \Omega \) fixed, note that \(v_i:=u_i-z\) both satisfy the deterministic ODE (3.9) with respective initial data \(v_{i,0}=u_{i,0}\). In particular, for given \(\omega \), the a priori bounds (3.12) hold and \(u_i=v_i+z\) can be bounded as
for a common constant \(C_{\omega ,T}\) possibly depending on \(\omega \), T, and \(\Vert u_{i,0}\Vert \) only. Moreover, setting
we see that \(w=u_1-u_2\) satisfies, again for fixed \(\omega \), the deterministic ODE
Because \(u_i^\omega (t)\) are bounded a priori and F is locally Lipschitz on the finite-dimensional space \(E_N\), we have
for some local Lipschitz constant \(L_{\omega ,T}\) depending only on the above upper bound \(C_{\omega ,T}\) on \(\Vert u_i(\omega ,t)\Vert \). Grönwall’s inequality gives then
In particular if \(u_{1,0}=u_{2,0}\) then for a.a. \(\omega \in \Omega \) the paths \(u_1(\omega ,t)=u_2(\omega ,t)\) for all times. Because the local Lipschitz constant only depends on the initial data through their norms, this also proves the pathwise continuity with respect to the initial data, Definition 3.1(ii).
Finally, since by construction z is adapted to \(\mathcal {F}_t\) and v is obtained by a fixed point argument, it follows that the stochastic process u is adapted to the \(\sigma (u_0,\mathcal {F}_t)\) and the proof is complete. \(\square \)
We define
the derivative of E, M in the direction of the dissipation operator. Given our current choice (3.2) for \(\mathcal {L}\), these read explicitly
and
Clearly \(\mathcal {M}\) is nicely coercive, but will fail to give sufficient \(H^s\) control. The next step is to check that \(\mathcal {E}\) yields enough coercivity to get such a control. To this end, fix some \(\delta >0\) small enough so that \(s-\delta >d/2\) and \(\delta <\alpha \): By Lemma A.1 we have
and therefore
Interpolating \(\Vert u\Vert _{\dot{H}^{s-\delta }}\le \Vert u\Vert _{\dot{H}^{\alpha }}^{\theta }\Vert u\Vert _{\dot{H}^s}^{1-\theta }\) (for some suitable \(\theta \in (0,1)\) depending on \(\delta ,s,\alpha \)), and leveraging our growth condition (3.3), it is not too difficult to reabsorb the last exponential term into the first coercive ones, thus
for some small \(c>0\) and large \(C>0\) only depending on the various regularity parameters. In the same spirit, and since G grows exponentially, it is easy to check that \(\Vert u\Vert ^2\le C[1+ G(\Vert u\Vert )\Vert u\Vert ^2]\) for some structural constant C only depending on G, and in particular
Hence \(\mathcal {E},\mathcal {M}\) are both coercive. These will be the two key ingredients in controlling the compactness of solutions in some suitable sense, first by taking the inviscid limit \(\sigma \rightarrow 0\), and then \(N\rightarrow \infty \).
Let
We assume that the Brownian coefficients \(\{a_m\}_{m\in \mathbb {Z}}\) in (3.1) decay sufficiently rapidly so as to guarantee
which is an intrinsic decay condition on the noise. The first \(\alpha \)-decay appears of course due to the \((-\Delta )^\alpha \) dispersion in (1.1), and the \(\frac{d-1}{2}\) threshold appears for purely technical reasons related to the Sobolev embedding.
Proposition 3.3
Let \(u_0\) be a random variable in \(E_N\) independent of \(\mathcal {F}_t\) and such that \({\mathbb {E}}M(u_0)<\infty \). Let u be the solution to (3.4) with initial data \(u_0\). Then, we have
Proof
Itô’s formula readily gives
Since \(M'(u;h)=(u,h)\) and \(\left( u, -i \left[ (-\Delta )^{\alpha } u + P_N \left( 2\beta u e^{\beta |u|^2} \right) \right] \right) =0\), we have
and the result follows from \(M'' (u; e_m ,e_m) =1\) after taking expectations. \(\square \)
Proposition 3.4
Let u be the solution to (3.4) with initial data \(u_0\), a \(E_N\)-valued random variable independent of \(\mathcal {F}_t\). If \({\mathbb {E}}\, E(u_0)<\infty \), then
with \(C=C_M(4\beta ^2+2\beta )\) for some \(C_M>0\) depending on the manifold M only.
Proof
Owing to the conservation law \(E' \left( u; i \left[ (-\Delta )^{\alpha } u + P_N \left( 2 \beta u e^{\beta |u|^2} \right) \right] \right) =0\), we have by Itô’s formula
and thus
By definition (1.2) of the energy, and owing to the universal estimate \(\Vert e_m\Vert _{L^\infty }\le C \lambda _m^{\frac{d-1}{4}}\) (for some C depending on the manifold M only, see [9]) we have
and the result follows by definition of \(A^s_{N}\) with \(s=\alpha ,\frac{d-1}{2}\). \(\square \)
For fixed \(\sigma >0,N\in \mathbb {N}\), and given \(w\in L^2\) as well as a Borel set \(\Gamma \) of \(L^2(M)\), we define the transition kernel
where \(u(t;P_Nv)\) is the solution of the SDE (3.4) arising from the deterministic initial datum \(u(0)=P_Nv\) (constructed in Proposition 3.2). We define the semigroup \(\mathfrak {B}_{\sigma ,N}^t : L^\infty (L^2 ; \mathbb {R}) \rightarrow L^\infty (L^2 ; \mathbb {R})\) as
Owing to the pathwise continuity (ii) in Definition 3.1, we see that \(\mathfrak {B}_{\sigma ,N}^t\) actually satisfies the Feller property: For any fixed time, \(\mathfrak {B}_{\sigma ,N}^t\) maps \(C_b (L^2)\) to itself (the space of real-valued, bounded continuous functions on \(L^2\)). The dual semigroup \(\mathfrak {B}_{\sigma ,N}^{t*}: \mathfrak {p} (L^2 ) \rightarrow \mathfrak {p}(L^2)\) on probability measures \(\mathfrak {p}(L^2)\) over \(L^2(M)\) is accordingly defined as
By definition we have, for all \(\lambda \in \mathfrak {p}(L^2)\) and \(f\in C_b(L^2)\),
where u(t) is the unique solution of (3.4) emanating from the initial \(E_N\)-valued random variable \(u_0\) distributed according to \(\lambda \).
Theorem 3.5
Fix \(\sigma \in (0,1)\) and \(s>d/2\). For any \(N\ge 2\) there exists a probability measure \(\mu ^s_{\sigma ,N}\in \mathfrak {p}(L^2)\) which is stationary for (3.4) and concentrated on \(H^s\). Moreover, we have
and
for some \(C>0\) depending only on s, \(\alpha \), M but not on \(\sigma \) and N.
Proof
For \(t\ge 1\) we denote
Taking \(u_0\equiv 0\) and discarding the term \({\mathbb {E}}M(u(t))\ge 0\) in Proposition 3.3, we have by the characterization (3.17) that
From this and (3.15) we immediately infer
thus by Chebyshev’s inequality \(\lambda _t(B_R^c(H^{s-\alpha }))\le \dfrac{C}{R^2}\) uniformly in t. By the Sobolev embedding \(B_R(H^{s-\alpha })\) is compact in \(L^2\), hence the family \((\lambda _t)_{t\ge 1}\subset \mathfrak {p}(L^2)\) is tight. By the Prokhorov theorem the weak-\(*\) limit \(\mu ^s_{\sigma ,N}= \lim \limits _{n\rightarrow \infty }\lambda _{t_n}\) exists at least for a discrete sequence of times \(t_n\rightarrow +\infty \). Since \(\mathfrak {B}_{\sigma ,N}^{t*}\) is Feller, a standard Bogoliubov-Krylov argument (Lemma A.6 in the Appendix) ensures that \(\mu ^s_{\sigma ,N}\) is stationary for (3.4).
Observe that \(w\mapsto \Vert w\Vert ^2_{H^{s-\alpha }}\) is lower semicontinuous for the \(L^2\) topology, hence by (3.20) and standard lower semi-continuity of integral functionals (Lemma A.5 in the Appendix) we obtain
Taking an initial random variable distributed as \(u_0\sim \mu ^s_{\sigma ,N}\), we see that \({\mathbb {E}}(M(u_0))={\mathbb {E}}(\frac{1}{2}\Vert u_0\Vert ^2)={\mathbb {E}}_{\mu ^s_{\sigma ,N}}(\frac{1}{2}\Vert w\Vert ^2)<+\infty \). Thus we can legitimately apply Proposition 3.3 and leverage the invariance of \(\mu ^s_{\sigma ,N}\) to conclude that
and (3.18) follows.
In order to estimate now \(\mathcal {E}\), we first observe that if \(\Lambda >0\) is large enough in (3.3), the Sobolev embedding gives \(G(\Vert u\Vert _{H^{s}})=c\Vert u\Vert _{H^{s}}^2e^{\beta \Lambda \Vert u\Vert _{H^{s}}^2}\ge c \Vert u\Vert ^2 e^{2\beta \Vert u\Vert _{L^\infty }^2}\). We deduce from (3.13) that
and as a consequence
for some C only depending on the parameters. Since we just established (3.18), we see that
Because \(E_N\) is finite-dimensional we also have
whence by definition of the energy
This regularity allows now to take \(u_0\sim \mu ^s_{\sigma ,N}\) in Proposition 3.4: Omitting the details, the stationarity gives exactly as before
where the second inequality is a consequence of (3.22). This concludes the proof. \(\square \)
Recalling that \(\phi _N^t : E_N \rightarrow E_N\) is the global flow associated with (2.1), we define the corresponding flow on functions \(\Phi _N^t : C_b (L^2) \rightarrow C_b (L^2)\)
and its dual \(\Phi _N^{t *} : \mathfrak {p} (L^2) \rightarrow \mathfrak {p} (L^2)\)
Recalling from (3.15) that \(\mathcal {M}(u)\) controls \(\Vert u\Vert ^2_{H^{s-\alpha }}\), and due to the compact embedding \(H^{s-\alpha }\subset \subset L^2\), (3.18) shows that the family \(\Big \{\mu _{\sigma ,N}^s\Big \}_{\sigma \in (0,1)}\subset \mathfrak {p}(L^2)\) is tight. By Prokhorov’s theorem there exists a weak limit
at least for a discrete sequence \(\sigma _k\rightarrow 0^+\). For the ease of notations we will simply denote \(\mu ^s_{k ,N}=\mu ^s_{\sigma _k ,N}\), \(\mathfrak {B}_{k,N}^{t}=\mathfrak {B}_{\sigma _k,N}^{t}\), \(\mathfrak {B}_{k,N}^{t*}=\mathfrak {B}_{\sigma _k,N}^{t*}\) and so on along this discrete sequence.
Proposition 3.6
The measure \(\mu _N^s\) is invariant under \(\phi _N^t\) and satisfies
and
where \(C>0\) only depends on \(\alpha ,s,\beta ,M\) but not on N.
Proof
Since \(\mu ^s_N\) and \(\mu ^s_{\sigma ,N}\) are supported on \(E_N\), we are actually working on a finite dimensional space and the functions \(\mathcal {E}\) and \(\mathcal {M}\) are thus continuous. We have moreover \(\mathcal {M}\ge 0\), and \(\mathcal {E}\) is lower bounded according to (3.14). As a consequence we can apply Lemma A.5, and our estimates (3.24) and (3.25) follow from \(\mu ^s_{k,N}\rightharpoonup \mu ^s_N\) in (3.19) and (3.18).
Equality in (3.23) is more subtle, as \(\mathcal {M}(u)\) is unbounded and the weak convergence is thus not enough to conclude that \({\mathbb {E}}_{\mu _{N}^s}\mathcal {M}(u)=\lim {\mathbb {E}}_{\mu _{k,N}^s}\mathcal {M}(u)\). By (3.18) we have first
Using (3.25), we get moreover
and (3.23) simply follows by taking first \(k\rightarrow +\infty \) and then \(R\rightarrow +\infty \).
Next, let us show that \(\mu _N^s\) is invariant under \(\phi _N^t\). Without loss of generality, we can consider \(t>0\) only. Recall that \(\mu ^s_{k,N}=\mathfrak {B}_{k,N}^{t*}\mu ^s_{k,N}\) is stationary, and that \(\mu ^s_{k,N}\) converges weakly to \(\mu _N\) in \(\mathfrak {p}(L^2)\). We will show below that \(\mathfrak {B}_{k,N}^{t *}\mu ^s_{k,N}\) converges weakly to \(\Phi _N^{t *}\mu ^s_N\), and the invariance of \(\mu ^s_N\) under \(\phi ^t_N\) will follow from
To this end, fix \(f\in C_b (L^2)\) and write \(M\ge 0\) for its uniform bound. Since all the measures involved at this stage are actually supported on the finite-dimensional space \(E_N\), we can argue with a slight abuse as if \(L^2(M)\) were finite-dimensional, and it is therefore not restrictive to assume that f is locally Lipschitz. (In a locally compact metric space the space of locally Lipschitz functions is dense in \(C_b\) for the uniform topology, see Lemma A.7 in the appendix.) We have
Using that \(\Phi _N^t\) is Feller we have \(\Phi _N^t f\in C_b(L^2)\), whence \(B \rightarrow 0\) as \(k\rightarrow \infty \). Next, let us deal with A. We fix a large R and write \(B_R=B_R(L^2)\). Since |f| is bounded by M, we have \(|\mathfrak {B}_{k,N}^t f|,|\Phi _N^t f|\le M\) as well and therefore
where the last inequality follows from (3.21) combined with Chebyshev’s inequality. For fixed \(\rho>0,t>0\) and any \(u_0\in B_R(L^2)\), let
where \(z_k\) is defined in (3.6) for \(\sigma =\sigma _k\) and \(u_k=u_k(t;P_Nu_0)\) is the unique global solution of the SDE (3.4) with deterministic initial datum \(P_Nu_0\). Let us emphasize that \(S_\rho \) depends on \(u_0\), which is here a fixed deterministic initial datum in \(B_R(L^2)\). We will prove shortly that, for any \(R>0,\rho >0\) and fixed \(t>0\),
Temporarily admitting this claim, let \(L_R\) be the Lipschitz constant of f on the ball \(B_R\). Using that f is \(L_R\)-Lipschitz on \(B_R\) and globally M-bounded, we have
Owing to our claim (3.27) we see that \(A_{1,1}\rightarrow 0\) as \(k\rightarrow \infty \) for fixed R. On the other hand, by Itô isometry we get for \(u_0\in B_R(L^2)\)
where the last inequality follows by Proposition 3.3 with \({\mathbb {E}}\Vert u_k(0)\Vert ^2=\Vert u_0\Vert ^2\le R^2\). Recalling (3.8), we also have \({\mathbb {E}}\sup \limits _{\tau \le t}\Vert z_k(\tau )\Vert ^2 \le C \sigma ^2_kt\). Thus, using Chebyshev’s inequality, we get in \(A_{1,2}\)
where C is independent of \(R,\rho ,k\). Summarizing, we have estimated, for any fixed \(t>0, R>0,\rho >0,k\in \mathbb {N},f\in C_b\cap Lip(L^2)\),
for some constants \(C>0\) independent of \(R,\rho ,k\). Taking first \(k\rightarrow \infty \) with \(A_{1,1}\rightarrow 0\) for fixed R, then \(\rho \rightarrow \infty \), and finally \(R\rightarrow \infty \), the result follows.
Finally, it remains to establish our claim (3.27). To simplify notations, for fixed \(u_0\in L^2\) we denote \(u=u(t;u_0)=\phi _N^t( P_N u_0)\) and \(v_k= v_k (t;u_0)\), where \(v_k\) is the unique solution to (3.9) with \(\sigma =\sigma _k\) and initial datum \(P_N u_0\). We set
and recall that, by construction, the unique solution to (3.4) is \(u_k(t;u_0)=v_k(t;u_0) +z_k(t)\) (see the proof of Proposition 3.2). Writing
and recalling from (3.8) that \({\mathbb {E}}\Vert z_k(\tau )\Vert ^2 \le 2\sigma _k^2t\rightarrow 0\) uniformly in \(\tau \in [0,t]\), our claim (3.27) amounts to showing that
We will first show that \(w_k(t)\rightarrow 0\) for \(\mathbb {P}\)-a.a. \(\omega \in \Omega \), and then upgrade to uniform convergence of the expected value by restricting to \(\omega \in S_\rho \).
Subtracting the two equations \(\partial _tu -\partial _t v_k\), some elementary algebra gives
with initial datum \(w_k (0)=0\). Recall from (3.11)(3.12) that, for fixed \(\omega \in \Omega \) and \(t>0\), \(\Vert v_k(\tau )\Vert ,\Vert z_k(\tau )\Vert \le C_{\omega ,t}\) uniformly in \(\tau \in [0,t]\) and \(k\rightarrow \infty \), hence \(\Vert u_k(\tau )\Vert =\Vert z_k(\tau )+v_k(\tau )\Vert \le C_{\omega ,t}\) as well. Since on the finite dimensional space \(E_N\) there holds \(\Vert u\Vert _{L^\infty (M)}\le C_N\Vert u\Vert _{L^2(M)}\), we see that, for a fixed \(\omega \), the functions \(u,u_k,v_k,z_k\) are bounded in \(L^\infty ((0,t)\times M)\) uniformly in k. We write \(\Vert \cdot \Vert _{L^\infty _{tx}}\) for this norm. We have moreover
where \(f_{2p}\) and \({\tilde{f}}_{2p}\) are explicit polynomials of degree 2p. It is not difficult to check that the series \(F(u,v)=\sum _{p\ge 0} f_{2p}(u, v)\) and \({\tilde{F}}(v,z)= \sum _{p\ge 0} {\tilde{f}}_{2p}(v,z) \) are converging. By (3.28) we find in any fixed time-interval \(\tau \in [0,t]\)
Observe that, by definition (3.26) of \(S_\rho \), we have \(\sup \limits _{\tau \in [0,t]}\Vert z_k(\tau )\Vert \le \rho \sigma _k t\) for \(\omega \in S_\rho \). As a consequence, for \(\omega \in S_\rho \) the upper bound \(Z_{\omega ,t}\) in (3.11) can be taken of the form
and thus from (3.12) with a deterministic initial datum \(v_k(0)=P_Nu_0\in B_R(L^2)\) we control
uniformly in \(\omega \in S_\rho \) and \(u_0\in B_R\). With these uniform bounds on \(z_k,v_k\) and \(u_k=v_k+z_k\), an easy application of Young’s inequality in (3.29) immediately gives
where \(C_1,C_2\) depend on \(\rho ,R,t\) but not on \(k,\omega \in S_\rho , u_0\in B_R\). Grönwall’s inequality with \(w_k (0)=0\) easily gives
uniformly in \(\omega \in \Omega \) and as a consequence
For a given \(u_0\in B_R\) we have \(\Vert w_k (t;u_0)\Vert 1_{S_\rho } \le \sup \limits _{{\tilde{u}}_0 \in B_R} \Vert w_k (t; {\tilde{u}}_0 )\Vert 1_{S_\rho }\), hence
and the proof is complete \(\square \)
3.2 The large dimension limit \(N\rightarrow \infty \)
Proposition 3.7
There exist a measure \(\mu ^s\in \mathfrak {p}(L^2)\) such that, up to some subsequence (not relabeled here),
Moreover, we have
and
for some C only depending on the various parameters.
Note that, for finite N, the measure \(\mu ^s_N\in \mathfrak {p}(L^2)\) is supported on the finite-dimensional space \(E_N\), and can therefore be viewed unambiguously as a probability measure over \(H^r\) for any \(r\ge 0\). For definiteness and consistency we consider the limit \(\mu ^s\) as a probability measure over \(L^2\) as well, but it is part of our statement that \(\mu ^s\) can also be considered as a probability measure on \(H^r\) for any \(r<s\).
Proof
From (3.24) and (3.14) we see that
and from (3.23) combined with (3.15)
Whence
uniformly in N. By Chebyshev’s inequality we have thus \(\mu ^s_N(B_R^c(H^s))\le \frac{C}{R^2}\), and since \(B_R(H^s)\) is compactly embedded in \(L^2\) we see that the sequence \(\{\mu ^s_N\}_{N\ge 1}\) is tight over \(L^2\). Prokhorov’s Theorem guarantees that, up to extraction of a subsequence if needed, \(\mu ^s_{N}\rightharpoonup \mu ^s\) for some weak limit \(\mu ^s\in \mathfrak {p}(L^2)\). Actually, for any \(r<s\) the embedding \(B_R(H^s)\) into \(H^r\) is compact, hence the very same argument shows that \(\mu ^s_{N}\) is weakly compact over \(H^r\) for any \(r<s\). By uniqueness of the limit we see that the weak convergence holds for any such r along this particular subsequence. For the remaining estimates one can proceed exactly as in the proof of Proposition 3.6, with the minor difference that the various functions \(\mathcal {M}(u),\Vert u\Vert ^2\dots \) appearing in the integrals are not continuous anymore but only lower semicontinuous (with respect to the \(L^2\) topology, which is enough in order to safely apply Lemma A.5). \(\square \)
Now that we have a natural candidate \(\mu ^s=\lim \mu ^s_N\) as a “good” probability measure for (1.1), it remains to construct global solutions and to show that \(\mu ^s\) is indeed invariant for the corresponding flow. Here we carry out most of the preparatory technical work.
A key issue will be to prove that the \(H^r\) norms grow slowly enough (logarithmically in time) for \(r<s\). In order to measure this growth in the correct time scale, let us introduce the monotone increasing function
For the reader’s convenience it is worth anticipating that the growth function \(\zeta \) in Theorem 1.1 will be \(\zeta =2{\tilde{G}}^{-1}\), roughly speaking.
Proposition 3.8
Let \(d/2< r<s\), \(N\ge 0\). For any \(i\in \mathbb {N}\) there exists a set \(\Sigma _{N,r}^i\subset E_N\) such that
for some \(C>0\) depending only s. Moreover for all \(u_0 \in \Sigma _{N,r}^i\), we have
One can think here of i as a fixed time delay. Given such a fixed delay, (3.34) means that there is a large set on which we have an explicit logarithmic \(H^r\) control, and this set is exponentially large for large delays.
Proof
For \(j\ge 1\), we set
a ball of radius \(i+j\) in the \({\tilde{G}}\) scale. It may help to think of i as a fixed time delay appearing in (3.34), while \(j\in \mathbb {N}\) can be thought of as a discrete “current running time”.
Let \(T=T(s,R)\) be the local time of existence from Proposition 2.1 with \(R=\tilde{G}^{-1} (i+j)\), and
On can think of this as the set of initial data allowing to maintain a logarithmic control up to large times \(|t|\lesssim e^j\) and modulo a fixed delay i. Using the invariance of \(\mu _N^s\), we get
We claim now that
for some C depending only on r, s. Indeed, we first observe from \(\int _M |u|^2e^{\beta |u|^2}\ge \Vert u\Vert ^2\) in (3.14) that
Note that, in dimension \(d\ge 2\) and because we fixed \(\alpha \le 1\) with \(d/2<r<s\), there holds \(\alpha<r<s\). By standard interpolation there exists \(\theta \in (0,1)\) depending on r, s, and \(\alpha \) such that \(\Vert u\Vert _{H^r}\le \Vert u\Vert ^\theta _{H^\alpha } \Vert u\Vert _{H^s}^{1-\theta }\). Moreover, owing to the exponential growth of G, there exists R depending only on \(G,\theta \) such that the function \(\rho \mapsto G(\rho )/\rho ^{2\frac{1-\theta }{\theta }}\) is monotone increasing at least for large \(\rho \ge R\ge 1\). Thus if \(\Vert u\Vert _{H^r} \ge R\ge 1\) we have
On the other hand, since \(\mathcal {E}(u)\ge -C\) from (3.14), for \(\Vert u\Vert _{H^r}\le R\) we have trivially have \(G(\Vert u\Vert _{H^r})\le G(R)=C\le C(1+\mathcal {E}(u))\) and our claim (3.36) follows.
Back to our main argument: From (3.24) and (3.36) we have therefore
uniformly in N, and Chebyshev’s inequality yields
(recall that we defined \(k_j=\left\lfloor e^j/T \right\rfloor \)). By (2.2) we have \(T=T(s,R)=\frac{c(s)}{\beta e^{C(s)\beta R^2}}\) with here \(R=2{\tilde{G}}^{-1}(i+j)\), hence
The last inequality follows from our growth condition (3.3), which gives explicitly here \({\tilde{G}}^{-1}(z)\sim \sqrt{z/\Lambda \beta }\) for large z and therefore \(e^{-C\beta [2{\tilde{G}}^{-1}(i+j)]^2}\sim e^{-\frac{4C}{\Lambda }(i+j)}\ge e^{-(i+j)/2}\) if \(\Lambda \ge 8C(s)\) is chosen large enough.
Set now
In words, \(\Sigma _{N,r}^i\) is the set of initial data that maintain logarithmic control for all times (here for \(|t|\le e^j\) for all \(j\in \mathbb {N}\)), up to some fixed delay i and measured in the \(H^r\) scale. Our claim (3.33) finally follows from the summability
It remains now to establish (3.34). Fix first \(i,j\in \mathbb {N}\). We claim that for any \(u_0 \in \Sigma _{N,r}^{i,j}\) there holds
Indeed for \(t\le e^j\), we can write uniquely \(t=kT +\tau \) where \(k\le k_j=\left\lfloor e^j/T \right\rfloor \) is an integer and \(\tau \in [0,T)\). As a consequence \(\phi ^t_N (u_0)= \phi ^{\tau }_N( \phi ^{kT}_N (u_0)) = \phi ^\tau _N (w)\), with \(w=\phi ^{kT}_N (u_0)\in B_{N,r}^{i,j}=B_R(H^r)\) by definition (3.35) of \(\Sigma _{N,r}^{i,j}\) and with \(R={\tilde{G}}^{-1}(i+j)\). Since \(\tau \le T\), Proposition 2.1 yields \(\Vert \phi ^\tau _{N}(w)\Vert _{H^r}\le 2R\) as claimed.
Fix now \(i\in \mathbb {N}\) and \(u_0\in \Sigma _{r,N}^i\), and choose an arbitrary \(t\ge 0\). There exists a unique \(j\ge 1\) such that \(e^{j-1}\le 1+t< e^j\), in particular \(j\le 1+\ln (1+t)\). By definition of \(\Sigma _{r,N}^i\) we have \(u_0\in \Sigma _{r,N}^{i,j}\) for this j in particular, hence from our previous claim
and the proof is complete. \(\square \)
The next result allows to trade off some Sobolev regularity for control over the time delay.
Lemma 3.9
Fix any \(d/2<r<s\), \(0<r_1 <r\), and \(t\in \mathbb {R}\). There exists \(i_1=i_1(t) \in \mathbb {N}^*\) such that, if \(u_0 \in \Sigma _{N,r}^i\) for some i, then \(\phi _N^t (u_0) \in \Sigma _{N,r_1}^{i+i_1}\).
Proof
Assume without loss of generality that \(t>0\), and let \(u_0 \in \Sigma _{N,r}^i\) for some i. Then, as already established in (3.38), we have for any \(j\ge 1\)
Let \(i_1=i_1 (t)\) be such that \(e^j +t \le e^{j+i_1}\) for all \(j\ge 1\). Again, we get
For every \(u_0 \in \Sigma _{N,r}^i\) we have by definition \(\Vert u_0\Vert \le \Vert u_0\Vert _{H^r} \le 2 \tilde{G}^{-1} (1+i)\), and since mass is preserved
By interpolation, for any \(r_1 \in (0, r)\) there is \(\theta \in (0,1)\) depending only on \(r,r_1\) such that
for all j and \(t_1\le e^j\). This is exactly the definition of \(\phi _N^{t+t_1} (u_0)\in \Sigma ^{i+i_1}_{N,r_1}\), and the proof is complete. \(\square \)
We set
and
In words, \(\Sigma _r\) is the set of initial data that maintain \(H^r\) logarithmic control for all times, up to some delay i.
Proposition 3.10
Let \(\mu ^s=\lim \mu ^s_{N}\) be the probability measure constructed in Proposition 3.7. Then
Proof
Using the Portmanteau theorem, the definition (3.37) of \(\Sigma _{N,r}^i\) and the fact that \(\Sigma _{N,r}^i \subset \Sigma _r^i\), we have
where we crucially used (3.33) for the last inequality. Using now that \(\overline{\Sigma _r^i}\) is non-decreasing in i, we get
and \(\Sigma _r\) has thus full \(\mu ^s\) measure as claimed. \(\square \)
We are now ready for the core of the matter, namely the construction of global solutions.
3.3 Global well-posedness and variations
Proposition 3.11
Let \(d/2<r <s\). For \(\mu ^s\)-almost all \(u_0\in \Sigma _r\) there is a unique global in time solution \(\phi ^t(u_0)\) to (1.1). This solution satisfies moreover
for some \(i=i(u_0,r)\).
Proof
Recalling that \(\mathcal {M}(u)\) controls \(\Vert u\Vert ^2\), we see from (3.30) and (3.31) that
and thus \(\mu ^s\)-almost all initial data \(u_0\in \Sigma _r\subset L^2\) actually lie in \(H^s\). By definition (3.40) of \(\Sigma _r\) we have that \(u_0 \in \overline{\Sigma _r^i}\) for some \(i\in \mathbb {N}\). Assume first that \(u_0\in \Sigma _r^i\). By definition (3.39) of \(\Sigma _r^i\), there exists a sequence \(u_{0,N} \in \Sigma _{N,r}^i\) converging to \(u_0\) in \(H^r\) as \(N\rightarrow \infty \). Our previous estimate (3.34) readily gives
In particular for \(t=0\) we have \(\Vert u_{0,N}\Vert _{H^r} \le 2 \tilde{G}^{-1} (1+i)\), whence taking the limit in N
The same holds if \(u_0 \in \partial \Sigma _r^i\), simply taking a sequence \(u_{0,k}\rightarrow u_0\) in \(H^r\) for some sequence \(u_{0,k}\in \Sigma _r^i\) satisfying the same estimate.
Fix an arbitrary \(T_0>0\). Clearly it suffices to prove that \(\phi ^t(u_0)\) is well-defined and satisfies (3.41) at least for times \(|t|\le T_0\). To see this, set \(R=2 \tilde{G}^{-1} (1+i +\ln (1+T_0)) +1\ge \Vert u_0\Vert _{H^r}+1\), and let \(T=T(r,2R)\) be the common local existence time from Propositions 2.1 and 2.3. If \(T\ge T_0\) then Lemma 2.4 allows passing to the limit in (3.42) for all \(|t|\le T_0\) and our claim follows. Assume otherwise that \(T<T_0\). Consider first \(u_0 \in \Sigma _r^i\), and let \(u_{0,N} \in \Sigma _{N,r}^i\) be the corresponding sequence converging to \(u_0\) in \(H^r\). By Lemma 2.4 we have thus
Observe from (3.42) that, for \(|t|\le T< T_0\) we have \(\Vert \phi ^t_N(u_{0,N})\Vert \le 2{\tilde{G}}^{-1}(1+i+\ln (1+|T|))\le R-1\). The triangle inequality thus gives for \(|t|\le T\) and large N
and as a consequence \(\phi ^{T}(u_0)\in B_R(H^r)\). Bootstrapping and exhausting \(t\in [-T_0,T_0]\) by finitely many intervals of fixed size T, we can repeat the argument as many times as needed and the claim follows.
Finally, if \(u_0 \in \partial \Sigma _r^i\), let \(u_{0,k}\in \Sigma _r^i\) be a sequence converging to \(u_0\) in \(H^r\). For the same R, T depending only on the arbitrary choice of \(T_0\) and the Sobolev exponent r, we have \(\Vert u_0\Vert _{H^r}\le R-1\) hence clearly \(u_0,u_{0,k}\in B_R(H^r)\) at least for large k. By Lemma 2.4 we see that for \(|t|\le T=T(r,2R)\) there holds
whence by triangular inequality and for all times \(|t|\le T\)
The result then follows by the same bootstrap argument as before and we omit the details. \(\square \)
Let \((s_n)_n\) be an increasing sequence such that \(\lim \limits _{n\rightarrow \infty } s_n =s\), and set
Proposition 3.12
\(\Sigma ^s\) has full \(\mu ^s\)-measure. Moreover, up to removing a \(\mu ^s\)-null set from \(\Sigma ^s\) if needed, for any \(u_0\in \Sigma ^s\) there is a unique global solution to (1.1) and the corresponding global flow satisfies \(\phi ^t (\Sigma ^s )=\Sigma ^s\) for all \(t\in \mathbb {R}\).
Proof
Up to removing a null set from each \(\Sigma _{s_n}\) if needed, we can assume from Proposition 3.11 that a global solution \(\phi ^t (u_0)\) exists for any \(u_0\in \Sigma _{s_n}\), and thus for any \(u_0\in \Sigma ^s\). Since \(\Sigma _r\) has full measure for any \(r<s\) (Proposition 3.10) and since the intersection is countable, clearly \(\Sigma ^s\) has full measure.
Let \(u_0 \in \Sigma ^s\), and fix an arbitrary rank \(n\in \mathbb {N}\). We will show that \(\phi ^t(u_0)\in \Sigma _{s_m}\) for all \(m\le n-1\), hence \(\phi ^t(u_0)\in \Sigma ^s\). The key point will be the cascade of regularity from Lemma 3.9, allowing to propagate \(H^{s_n}\) estimates to \(H^{r}\) control for all \(r<s_n\). More precisely: By definition of \(\Sigma _{s_n}\) there exists \(i_n\ge 1\) such that \(u_0 \in \overline{\Sigma _{s_n}^{i_n}}\).
-
Assume first that \(u_0\in \Sigma _{s_n}^{i_n}\): by definition of \(\Sigma _{s_n}^{i_n}\) there exists a sequence \(\{u_{0,N}\}_{N}\subset \Sigma _{N,s_n}^{i_n}\) converging to \(u_0\) in \(H^{s_n}\) as \(N\rightarrow \infty \). According to Lemma 3.9 there exists \(I_n=I_n(t)\in \mathbb {N}\) such that \(\phi _N^t (u_{0,N}) \in \Sigma _{N, r}^{i_n+I_n}\) for all \(r<s_n\), in particular for \(r=s_1,\dots ,s_{n-1}\). Leveraging Lemma 2.4 and passing to the limit \(\phi ^t_N (u_{0,N})\rightarrow \phi ^t (u_0)\), we see that \(\phi ^t (u_0)\) lies in \(\Sigma _{r}^{i_n+I_n}\) for \(r=s_1,\dots ,s_{n-1}\).
-
If now \(u_0 \in \partial \Sigma _{s_n}^i\), there is a sequence \(\{u_{0,k}\}_k \subset \Sigma _{s_n}^{i_n}\) converging to \(u_0\) in \(H^{s_n}\). We actually just proved that \(\phi ^t (\Sigma _{s_n}^{i_n}) \subset \Sigma _{r}^{i_n +I_n}\) for all \(r<s_n\), hence \(\phi ^t(u_{0,k})\in \Sigma _{r}^{i_n +I_n}\) for such r. Since \(\phi ^t (.)\) is continuous (Lemma 2.4), we have \(\phi ^t (u_0) =\lim \limits _{k\rightarrow \infty } \phi ^t (u_{0,k}) \in \overline{\Sigma _{r}^{i_n+I_n}}\) for all \(r<s_n\), in particular for \(r=s_1,\dots ,s_{n-1}\).
In any case we see that \(\phi ^t (u_0)\in \cap _{m\le n-1}\Sigma _{s_m}\). Because \(n\in \mathbb {N}\) was arbitrary we conclude that \(\phi ^t(u_0)\in \left( \cap _{n\ge 0}\Sigma _{s_n}\right) =\Sigma ^s\), hence \(\phi ^t(\Sigma ^s) \subset \Sigma ^s\).
Conversely, let \(u\in \Sigma ^s\). Letting \(u_0 = \phi ^{-t} (u)\), we have from the previous step that \(\phi ^{-t}(\Sigma ^s)\subset \Sigma \) and thus \(u_0\in \Sigma ^s\). Thus \(u=\phi ^t (u_0)\in \phi ^t(\Sigma ^s)\) and therefore \(\Sigma ^s\subset \phi ^t(\Sigma ^s)\) as well. \(\square \)
Proposition 3.13
The measure \(\mu ^s\in \mathfrak {p}(L^2)\) is invariant under \(\phi ^t\) on \(\Sigma ^s\), in the sense that \(\mu (\phi ^{-t}(A))=\mu (A)\) for any \(r<s\) and all \(H^r\)-measurable set \(A\subset \Sigma ^s\).
Proof
Note that, since \(\Sigma ^s\subset H^{s}\) has full measure and \(\phi ^t(u)\) is globally defined for all \(u\in \Sigma ^s\), the flow \(\phi ^t(u)\) is unambiguously defined for \(\mu ^s\)-a.e. \(u\in L^2\) and therefore it suffices to prove that \(\int _{H^r}f(u)\mu ^s(du)=\int _{H^r} f(\phi ^t(u))\mu ^s(du)\) for all \(f\in C_b(H^r)\). (Here we choose unambiguously to view \(\mu ^s\) as a probability measure over \(H^r\), for any fixed \(r<s\).) In fact, by Ulam’s tightness theorem, it suffices to prove that
for all \(f\in C_b(H^r)\) and any \(H^r\)-compact set K. As in the proof of Proposition 3.6, and because we already know that \(\mu ^s_N\) is invariant under \(\phi ^t_N\) (Proposition 3.6) and converges weakly to \(\mu ^s\) (Proposition 3.7), we have
and the statement will follow if we can prove that
(up to subsequences, not relabeled here for the ease of notations).
Clearly, by repeated iterations if needed, it is enough tho establish the statement for \(|t|\le T\) small enough. Fix a large \(R>0\) so that \(K\subset B_B(H^r)\), let \(T=T(r,R)\) as usual, and pick any \(|t|\le T\). We write then
As \(\mu ^s_N\rightharpoonup \mu ^s\) weakly over \(H^r\) and \(f\circ \phi ^t\in C_b(H^r)\) for fixed \(|t|\le T\) (Lemma 2.4), clearly the second term \(B\rightarrow 0\). Because K is compact and according to Lemma A.7, we can assume without loss of generality that \(f\in C_b(H^r)\) is actually L-Lipschitz on K for some L. We can therefore estimate for the A term
Our Corollary 2.5 guarantees that this latter quantity converges to zero for fixed \(|t|\le T\), and the proof is complete. \(\square \)
We can now fully establish our main result for high regularities
Proof of Theorem 1.1
Recall that, for fixed Brownian coefficients \((a_m)_m\) in (3.1), we constructed \(\mu ^s\) so that \({\mathbb {E}}_{\mu ^s} \mathcal {M}(u)=\frac{A^0}{2}\) from Proposition 3.7. Scaling \((a_m)_m\rightarrow (\sqrt{n}a_m)_m\) for an arbitrary \(n\in \mathbb {N}\), all the quantities \(A^{r}=\sum _m\lambda _m^r|a_m|^2\) scale as \(A^r\rightarrow nA^r\) for any fixed r and we can therefore choose the \(a_m\) coefficients so that in particular \(A^0=n\) for any given n. According to Proposition 3.7 the corresponding stationary measure \(\mu ^{s,n}\) satisfies \({\mathbb {E}}_{\mu ^{s,n}} \,\mathcal {M}(u)=n/2\). Let \(\Sigma ^{s,n}\) be the corresponding invariant set constructed in Proposition 3.12. Owing to \(\mathcal {M}(u)=\Vert u\Vert ^2_{\dot{H}^{s-\alpha }}+ G(\Vert u\Vert _{H^s})\Vert u\Vert ^2\le \Vert u\Vert _{H^s}^2+G(\Vert u\Vert _{H^s})\Vert u\Vert _{H^s}^2\), we see that \(\Sigma ^{s,n}\) contains data of size at least \(\Vert u\Vert _{H^s}\ge \xi (n/2)\), where \(\xi \) is the inverse of the increasing map \(\rho \mapsto \rho ^2+G(\rho )\rho ^2\).
Let us now define
Clearly \(\Sigma ^{s}\) has full-measure (because \(\mu ^{s,n}(\Sigma ^{s,n})=1\)), is contained in \(H^s\), and \(\phi ^t\) is globally well-defined on \(\Sigma ^s\). Because \(\Sigma ^{s,n}\) contains large data \(\Vert u\Vert _{H^s}\ge \xi (n/2)\) for fixed n, clearly \(\Sigma ^{s}\) contains arbitrarily large data. Since all the \(\mu ^{s,n}\) are invariant under \(\phi ^t\), so is \(\mu ^{s}\).
From (3.30) and (3.31), for fixed n we have that
hence
Finally, pick an arbitrary \(u_0\in \Sigma ^{s}\) and fix \(r<s\). By definition we have \(u_0\in \Sigma ^{s,n}\) for some \(n\in \mathbb {N}\), hence from Proposition 3.11
for some i only depending on \(u_0\) and r. With our choice of \(G(\rho )=c\rho ^2e^{\Lambda \beta \rho ^2}\) it is easy to check that \({\tilde{G}}(\rho )=\ln (1+ G(\rho ))\sim \Lambda \beta \rho ^2\) is convex and superlinear at infinity, hence \(\zeta (z)=2 \tilde{G}^{-1}(z)\sim 2\sqrt{z/\Lambda \beta }\) is concave and sublinear at infinity. From there it is a simple exercise to show that
for some structural prefactor \(C_\zeta (\cdot )\) depending only on the growth parameters \(\Lambda ,\beta \), and the proof is complete. \(\square \)
4 Low regularity
In this section we consider the case \(s\le d/2\). As already mentioned, the main obstacle in this case is the fact that the existence time for solutions to the projected Eq. (2.1) is exponentially small as \(N\rightarrow \infty \), which is reminiscent from the fact that even the local well-posedness for the unprojected Eq. (1.1) is delicate, to say the least. As a consequence we will lose any quantitative control on the growth of \(H^r\) norms over time. In order to circumvent this difficulty we still implement a fluctuation-dissipation approach, but with a modified dissipation operator \(\mathcal {L}(u)\) in (1.5). Roughly speaking, the new dissipation will allow to gain an extra coercivity and control the exponential nonlinearity (which was previously achieved through a control on the \(H^s\) norm and the Sobolev embedding). For technical reasons, this will give a restriction on \(s\le \alpha +1\).
4.1 Compactness estimates and stationary measure
In this section, we take
Compared to our previous choice (3.2) for high regularities, the energy and mass dissipation \(\mathcal {E}(u)=E'(u;\mathcal {L}(u))\), \(\mathcal {M}(u)=M'(u;\mathcal {L}(u))\) read now
and
To deal with the mixed terms appearing in \(\mathcal {E}(u)\), we use an exponential Córdoba-Córdoba inequality (Lemma A.3 in the Appendix with parameter \(\gamma =s-\alpha \le 1\)) to control (for \(u\in E_N\) hence \((-\Delta )^\theta u\in E_N\) for all \(\theta \))
and
Discarding these nonnegative terms in (4.2), we obtain the coercivity
Remark 4.1
We stress that in the Córdoba-Córdoba inequality it is crucial that the exponent \(\gamma \le 1\), and this is the only reason why we had to restrict to \(s-\alpha \le 1\) in this section. For \(\alpha +1<s\le d/2\) we did not manage to control these two crossed terms, at least not for this choice of dissipation \(\mathcal {L}\), and the functionals \(\mathcal {M},\mathcal {E}\) do not provide sufficient control on the nonlinearity for the subsequent analysis to carry over.
At this point, and just as before in the high regularity regime \(s>d/2\), one should establish the stochastic well-posedness of
in the sense of Definition 3.1. Just as in Sect. 3.1 and Proposition 3.2, we construct the unique solution \(u=v+z\), with z still given by (3.6) and containing all the Brownian fluctuations. Similarly to (3.9), the corrector v should now solve, for any realization of \(u_0=u_0^\omega \) and \(z(t)=z^\omega (t)\), the deterministic problem
Standard Cauchy-Lipschitz arguments in the finite-dimensional space \(E_N\) guarantee local well-posedness, the challenge here is rather proving that solutions are actually global-in-time. We have
Using as before that \(\left\langle u,iue^{\beta |u|^2}\right\rangle =0\) (with \(u=v+z\)) in order to substitute \(-z\) for v in the very first term, this gives
In order to fully exploit the coercivity from the second \(-\sigma ^2\) term in the right-hand side, let \(\Phi _\sigma \) be the convex function given by Lemma A.4 with \(b=\beta \) and \(c=\sigma ^2\). Applying the generalized Young inequality (with \(\Phi ^*_\sigma \) the Legendre transform of \(\Phi _\sigma \)), we estimate
Recalling from (3.11) that \(\sup \limits _{\tau \le t}\Vert z^{\omega }(\tau )\Vert ^2\le \sigma ^2 C_{\omega ,t}\) for fixed \(\omega \in \Omega \), we have \(\Vert z^\omega (\tau )\Vert ^2_{L^\infty (M)}\le C_N \Vert z^\omega (t)\Vert ^2\le C_{N,\omega ,t} \sigma ^2\) for \(\tau \le t\) and we conclude that
Then either \(\Vert v(t)\Vert ^2_{H ^{s-\alpha }}(t)\le \Vert z(t)\Vert ^2_{H ^{s-\alpha }}\), or \(\frac{d}{dt}\frac{1}{2}\Vert v\Vert ^2 \le Vol(M) \Phi _\sigma ^*(C_{\beta ,\omega ,N,t}\sigma )\). Since z is bounded (for fixed \(\omega \)) both scenarios lead to some quantitative estimate of the form
preventing blow-up in finite time, and we conclude that v is indeed global-in-time for a.a. \(\omega \in \Omega \). Being the right-hand side of (4.6) a locally Lipschitz function of v, this a priori bound leads to continuity with respect to deterministic initial data, hence (4.6) is again stochastically globally well-posed in the sense of Definition 3.1.
Exactly as in the high regularity case, one can establish at this stage the equivalent of Proposition 3.3 and Proposition 3.4 (adapted to the new dispersion operator \(\mathcal {L}(u)\) in (4.1)). The coercivity (4.3) and (4.4) gives then enough compactness on the one-parameter family
and just like in the proof of Theorem 3.5 we conclude from a Bogoliubov-Krylov argument that
There exists a stationary measure \(\mu ^s_{\sigma ,N}\) for (4.5).
The next step is to take the inviscid limit \(\sigma \rightarrow 0\). This can be done here as well, and it is easy to show as in Sect. 3.1 that there exists \(\mu ^s_N\in \mathfrak {p}(L^2)\) such that, for a discrete sequence \(\sigma _k\rightarrow 0\)
Note however that, in order to prove that \(\mu ^s_N\) is invariant for the deterministic flow \(\phi _N^t\) in Proposition 3.6, we strongly exploited in (3.29) the fact that the bound (3.12) was uniform in \(\sigma \). For low regularities our corresponding bound (4.7) is unfortunately not uniform in \(\sigma \), and in fact blows-up as \(\sigma \rightarrow 0\) due to \(\Phi ^*_\sigma \) being the Legendre transform of \(\Phi _\sigma =\sigma ^2\Phi \) for a fixed \(\Phi \) (and as a consequence \(\Phi _\sigma ^*(z)=\sigma ^2\Phi ^*(z/\sigma ^2)\), which indeed blows-up because \(\Phi ^*\) grows super-exponentially at infinity but z is only controlled as \(\mathcal {O}(\sigma )\)).
Fortunately, it suffices to prove in fact that \(u_k=v_k+z_k\) in (3.29) remains bounded uniformly in \(\sigma _k\rightarrow 0\) for \(\omega \in S_\rho \) only (and not all \(\omega \in \Omega \) as in the proof of Proposition 3.6). The key idea for this is that, by definition (3.26) of \(S_\rho \), the martingale term is controlled uniformly in \(\omega ,\sigma \). More precisely, let \(u_k=u_k(t;P_Nu_0)\) be the solutions of the SDE (4.5) with dissipation \(\mathcal {L}(u)\) given by (4.1). As in the proof of Proposition 3.3, \(M=\frac{1}{2}\Vert u_k\Vert ^2\) has the Itô differential
Because \(\mathcal {M}\ge 0\), and since by definition (3.26) of \(S_\rho \) the martingale term is bounded by \(\rho \sigma _k t\) for \(\tau \le t\), we see that
for all \(\tau \le t\), uniformly in \(\sigma _k\rightarrow 0\) and \(\omega \in S_\rho \). The rest of the proof of Proposition 3.6 remains identical, and we conclude as before that
Moreover, and omitting the details for the sake of exposition, the coercivity (4.3) and (4.4) of \(\mathcal {M},\mathcal {E}\) give altogether
for some \(C>0\) independent of N. Compared to the high regularity in Sect. 3, we lost quantitative estimates on the growth-rate of \(H^r\) norms, but we gained in exchange \(L^2\) control over the projected nonlinearity.
4.2 Construction of global solutions
The goal is now to pass to the limit \(N\rightarrow \infty \), and this is where the Skorokhod-type argument by Burq et al. [4] comes into play. To this end, let \(u_{0}\) be an \(E_N\)-valued random variable distributed according to \(\mu ^s_N\in \mathfrak {p}(L^2)\) just constructed, and let \(u_N(t)=\phi ^t_N(u_0)\). As a first step we work in a finite time interval \(t\in [-T,T]\) for some fixed large \(T>0\). One can canonically consider the processes \((u_N (t))_{t\in [-T,T]}\) as a \(C([-T,T],H^{s-\alpha })\)-valued random variable. Let \(\nu ^s_{T,N}\) be its law. By invariance of \(\mu ^s_N\) we have
Consider the following functional spaces
and
and note that \(X_T^s\) is compactly embedded into \(Y_T^s\).
Proposition 4.2
We have
for some constant C independent of N.
Proof
From \(-i\partial _t u_N=(-\Delta )^\alpha u_N +2\beta P_N u_Ne^{\beta |u_N|^2}\) we have obviously
hence by (4.9) and the invariance of \(\mu ^s_N\)
By (4.8) this latter integral is bounded uniformly in N and the first part of our estimate follows.
In order to control now the \(L^1_tL^1_x\) norm of the nonlinearity, we first argue just as before to write
Let now \(\Phi (z)\) be the monotone increasing, convex function from Lemma A.4 with \(b=\beta \) and \(c=1\), defined implicitly as \(\Phi (ue^{\beta |u|^2})=u^2e^{\beta |u|^2}\). By Jensen’s inequality and (4.8) we have
uniformly in N, hence
and the proof is complete \(\square \)
Chebyshev’s inequality in (4.10) gives \(\nu ^s_{T,N}(B_R(X^s_T)^c)\le \frac{C}{R^2}\) uniformly in N, hence by the Prokhorov theorem there exists a weak limit
(up to a subsequence \(N_k\rightarrow \infty \), not relabeled here.) By lower semicontinuity in (4.10) we have moreover
By Skorokhod’s representation theorem, there exists a \(Y_T\)-valued stochastic process u distributed according to \(\nu ^{s}_T\in \mathfrak {p}(Y_T^s)\) and such that
Next, we want to show that u is a distributional solution of (1.1) for \(\mathbb {P}\)-a.a. \(\omega \). By construction \(u_N\) is a global distributional solution of (2.1), and the above almost sure convergence in \(C([0,T];H^{s-\alpha })\) implies the distributional convergence of the linear terms \(i\partial _t u_N-(-\Delta )^\alpha u_N\). Hence it clearly suffices to prove that the nonlinearity also passes to the limit in the sense of distributions,
To this end we decompose, for some arbitrarily large \(M\in \mathbb {N}\),
By the almost sure convergence of \(u_N\) to u and the continuity of \(F_M\), we see that the third term of the right-hand side converges almost surely to 0 at least in \(C ([-T,T] ; H^{s-\alpha })\). Let us now deal with the first term \(F_N(u_N)-F_N(u)\), and fix some large \(r>0\). Write for simplicity
so that \(F_N(u_N)=P_N v_N\) and \(F(u_N)=v_N\). By (4.11) we have, for almost-all \(\omega \in \Omega \), that
uniformly in N. If the Sobolev exponent \(-r<0\) is sufficiently negative, then there exists \({\varepsilon }={\varepsilon }(r,d)>0\) such that \(\Vert (1-P_N)f\Vert _{H^{-r}}\le CN^{-{\varepsilon }}\Vert f\Vert _{L^1}\) for all \(f\in L^1(M)\). As a consequence the first term in (4.13) can be controlled as
and in particular \(F_N(u_N)-F(u_N)\rightarrow 0\) in the distributional sense. The remaining terms \(F(u_N)-F_M(u_N)\) and \(F_M(u)-F(u)\) in (4.13) can be treated similarly, leveraging now a \(M^{-{\varepsilon }}\) decay rate for arbitrary M (here one should also exploit from (4.12) the information that \(\Vert ue^{\beta |u|^2}\Vert _{L^1_tL^1_x}\le C_\omega <\infty \) for a.a. \(\omega \)). This shows as desired that u is a distributional solution on \([-T,T]\).
In order to conclude the proof of Theorem 1.2 in the case \(s\le 1+\alpha \), observe that \(u=\lim u_N\) is a solution in the fixed, arbitrarily large time-interval \([-T,T]\). By a straightforward diagonal extraction argument, it is easy to recover as \(T\rightarrow \infty \) a limiting \(C(\mathbb {R};H^{s-\alpha })\)-valued stochastic process, still denoted \(u=\lim u_N\), that solves (1.1) globally in time. Denoting \(\nu ^s\) its law and passing to the limit in (4.9), we see that the law \(\nu ^s\big |_{t=t_0}\) of \(u(t_0)\) at an arbitrary time \(t_0\in \mathbb {R}\) is given by \(\mu ^s=\lim \mu ^s_N\). This means exactly that \(\mu ^s\) is invariant. Finally, the first term in (4.8) immediately passes to the limit as \(\int _{L^2}\Vert u\Vert ^2_{H^s}\mu ^s(du)\le \liminf \int _{L^2}\Vert u\Vert ^2_{H^s}\mu ^s_N(du)\le C<\infty \).
For the requirement that \(\mu ^{s}\) contains large initial data in Theorem 1.2, we argue exactly as in Sect. 3.3: we first retrieve \(\int _{L^2} \mathcal {E}(u)\mu ^s_N(du)=\frac{A^0_N}{2}\) as in Proposition 3.6, and then pass to the limit \(N\rightarrow \infty \) as in Proposition 3.7to retrieve
Scaling the Brownian coefficients \(a_m\rightarrow na_m\) for \(n\in \mathbb {N}\) gives a corresponding measure \(\mu ^{s,n}\in \mathfrak {p}(L^2)\) charging data of size \(\Vert u\Vert _{\dot{H}^s}\approx n\). The exact same construction (3.43) as before ultimately gives an invariant measure \(\mu ^{s}=\sum _n\frac{\mu ^{s,n}}{2^n}\) charging arbitrarily large data, in particular \(\mu ^{s}\) is not trivial and the proof of Theorem 1.2 is finally complete for \(s\le \alpha +1\).
References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer Science & Business Media, Berlin (2008)
Burq, N., Gérard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math. 126(3), 569–605 (2004)
Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166(1), 1–26 (1994)
Burq, N., Thomann, L., Tzvetkov, N.: Remarks on the Gibbs measures for nonlinear dispersive equations. Ann. Fac. Sci. Toulouse Math. 27(3), 527–597 (2018)
Cazenave, T.: Equations de Schrödinger non linéaires en dimension deux. Proc. R. Soc. Edinb. Sect. A 84(3–4), 327–346 (1979)
Córdoba, A., Córdoba, D.: A pointwise estimate for fractionary derivatives with applications to partial differential equations. Proc. Natl. Acad. Sci. 100(26), 15316–15317 (2003)
Constantin, P., Ignatova, M.: Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications. Int. Math. Res. Not. 2017(6), 1653–1673 (2017)
Colliander, J., Ibrahim, S., Majdoub, M., Masmoudi, N.: Energy critical NLS in two space dimensions. J. Hyperbolic Differ. Equ. 6(3), 549–575 (2009)
Donnelly, H.: Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds. J. Funct. Anal. 187(1), 247–261 (2001)
Földes, J., Sy, M.: Invariant measures and global well Posedness for the SQG equation. Arch. Ration. Mech. Anal. 241(1), 187–230 (2021)
Gutev, V.: Lipschitz extensions and approximations. J. Math. Anal. Appl. 491(1), 124242 (2020)
Ibrahim, S., Majdoub, M., Masmoudi, N., Nakanishi, K.: Scattering for the two-dimensional NLS with exponential nonlinearity. Nonlinearity 25(6), 1843–1849 (2012)
Kuksin, S., Shirikyan, A.: Randomly forced CGL equation: stationary measures and the inviscid limit. J. Phys. A 37(12), 3805–3822 (2004)
Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence, Volume 194 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2012)
Kuksin, S.B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115(1–2), 469–492 (2004)
Lam, F., Lippmann, B., Tappert, F.: Self-trapped laser beams in plasma. Phys. Fluids 20(7), 1176–1179 (1977)
Lebowitz, J.L., Rose, H.A., Speer, E.R.: Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys. 50(3–4), 657–687 (1988)
Miculescu, R.: Approximation of continuous functions by Lipschitz functions. Real Anal. Exchange, pp. 449–452 (2000)
Nakamura, M.: On nonlinear Schrödinger equations derived from the nonrelativistic limit of nonlinear Klein–Gordon equations in de Sitter spacetime. J. Differ. Equ. 259(7), 3366–3388 (2015)
Robert, T.: Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity. Preprint arXiv:2104.14348
Saanouni, T.: Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth. Bull. Belg. Math. Soc. Simon Stevin 17(3), 441–462 (2010)
Sy, M.: Almost sure global well-posedness for the energy supercritical Schrödinger equations. To appear in J. Math. Pures. Appl. arXiv:1905.03866
Sy, M., Yu, X.: Almost sure global well-posedness for the energy supercritical NLS on the unit ball of \(\mathbb{R}^{3}\). Preprint arXiv:2007.00766
Sy, M., Xueying, Y.: Global well-posedness and long-time behavior of the fractional NLS. Stoch PDE: Anal. Compt. (2021). https://doi.org/10.1007/s40072-021-00210-0
Mouhamadou, S.Y., Yu, X.: Global well-posedness for the cubic fractional NLS on the unit disk. Preprint arXiv:2011.05517
Sy, M.: Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation. Anal. PDE 11(8), 1841–1879 (2018)
Wang, B., Hudzik, H.: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equ. 232(1), 36–73 (2007)
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J.-B.C. supported by FCT - Fundação para a Ciência e a Tecnologia, under the Project: UIDB/04561/2020; L.M. was funded by the Portuguese Science Foundation through a personal grant 2020/00162/CEECIND as well as the FCT Project PTDC/MAT-STA/28812/2017.
A. Some technical results
A. Some technical results
Lemma A.1
Let \(s>d/2\). There exists a constant \(C=C(s,d)\) such that
and
Proof
First, let us recall that, for all \(u,v\in H^s\), we have
for some \(C=C(s)\). Using the previous inequality repeatedly, we get
and as a consequence we obtain as well
In order to estimate \(ue^{\beta |u|^2}-ve^{\beta |v|^2}\) we first exploit the algebraic identity \(|u|^{2k}u-|v|^{2k}v=(u-v)P_k(u,{\bar{u}},v,{\bar{v}})\), where \(P_k\) is a homogeneous polynomial of degree 2k defined recursively by \(P_{k}=|u|^{2k}+|v|^{2k}+uv P_{k-1}\) and \(P_0=1\). It is not difficult to check that \(\Vert P_k\Vert _{H^s}\le C^k (\Vert u\Vert ^2_{H^s}+\Vert v\Vert ^2_{H^s})^k\), hence
and the proof is complete. \(\square \)
Lemma A.2
(Córdoba-Córdoba inequality [6, 7]) Let \(\Phi \in C^2 (\mathbb {R},\mathbb {R})\) be a convex function satisfying \(\Phi (0)=0\), and fix \(\gamma \in (0,1]\). For any \(f\in C^\infty (M,\mathbb {R})\) there holds
Lemma A.3
For \(\gamma \in (0,1]\) and \(f\in C^{\infty }(M,\mathbb {C})\), we have
Proof
Let \(f=a+ib\). Applying twice the previous lemma with \(\Phi (r)=r^2/2\), separately to the real-valued functions a, b, we have first
Applying once again the Córdoba-Córdoba inequality to the real-valued function \(g=(a^2 +b^2)/2\) with this time \(\Phi (z)=e^z-1\), we find
pointwise on M. Finally multiplying by \(h=e^{(a^2+b^2)/2}\ge 0\) and integrating by parts in the right-hand side \(\int _M h(-\Delta )^\gamma h=\Vert (-\Delta )^{\gamma /2}h\Vert ^2=\Vert h\Vert ^2_{\dot{H}^\gamma }\) gives the result. \(\square \)
Lemma A.4
Let \(b,c>0\), and set \(f(u)=u e^{b u^2}\). Then the functional equation
admits a unique \(C^1\), strictly convex solution \(\Phi :\mathbb {R}^+\rightarrow \mathbb {R}^+\) given by
Proof
Observe that \(f:\mathbb {R}^+\rightarrow \mathbb {R}^+\) is \(C^\infty \), monotone increasing, bijective, and \(f'(u)\ge 1\). As a consequence \(f^{-1}:\mathbb {R}^+\rightarrow \mathbb {R}^+\) is also \(C^\infty \), monotone increasing, and bijective. Changing variables \(v=f(u)\Leftrightarrow u =f^{-1}(v)\), the functional equation obviously reads
The only thing left to check is that \(\Phi \) is convex. Differentiating twice \(\Phi ''(v)=c\frac{d^2}{dv^2}(f^{-1}(v)v)\), a straightforward but tedious computation leads ultimately to
with \(u=f^{-1}(v)\ge 0\). \(\square \)
Lemma A.5
([1, §5.1.1]) Let \((X,\textsf{d})\) be a Polish space and \(f:X\rightarrow \mathbb {R}\cup \{+\infty \}\) be a lower semicontinuous function bounded from below. Then the functional
is lower semi-continuous for the weak convergence of probability measures.
Lemma A.6
(Bogoliubov-Krylov, [24, Lemma B.1]) Let \((P_t)_{t\ge 0}\) be a Feller semi-group on a Banach space X and denote by \(P_t^*\) its adjoint. If there exists \(t_n \rightarrow \infty \) and \(\mu \in \mathfrak {p}(X)\) such that \(\frac{1}{t_n}\int _0^{t_n} P_t^*\delta _0 dt \rightharpoonup \mu \) in X then \(P_t^*\mu =\mu \) for all \(t\ge 0\).
Lemma A.7
([11, 18]) Let (X, d) be a compact metric space. Any \(f\in C(X;\mathbb {R})\) can be uniformly approximated by a sequence of globally Lipschitz functions.
Lemma A.8
(modified Arzelà-Ascoli) Let \((X,\textsf{d})\) be a Polish space, \(K\subset X\) a compact set, and \(f_n:K\subset X\rightarrow X\) an equicontinuous sequence of functions converging pointwise
Then
Proof
Fix \({\varepsilon }>0\). By equicontinuity there is \(\delta >0\) such that \(\textsf{d}(f_n(x),f_n(y))+\textsf{d}(f(x),f(y))\le {\varepsilon }/2\) as soon as \(\textsf{d}(x,y)\le \delta \). Take a finite cover \(K\subset \cup _i B(x_i,\delta )\) of radius \(\delta \). For \(x\in K\) we have by triangle inequality
for some suitable i such that \(x\in B(x_i,\delta )\), and the claim follows. \(\square \)
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Casteras, JB., Monsaingeon, L. Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity. Stoch PDE: Anal Comp 12, 416–465 (2024). https://doi.org/10.1007/s40072-023-00287-9
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DOI: https://doi.org/10.1007/s40072-023-00287-9