Absolutely continuous solutions for continuity equations in Hilbert spaces

Abstract

We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure γ which is Fomin-differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where γ is the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator Dx is closable with respect to L p (H, γ) and a recent formula for the commutator DxPt − PtDx where Pt is the transition semigroup corresponding to the reaction-diffusion equation, [DaDe14]. We stress that Pt is not necessarily symmetric in this case. This uniqueness result is an extension to such γ of that in [DaFlRoe14] where γ was the Gaussian invariant measure of a suitable Ornstein-Uhlenbeck process.Résumé. On démontre l'existence d'une solution de quelqueséquations de continuité dans un espace de Hilbert séparable. On s'interesse aux solutions absolument continues par rapportà une mesure de reference γ que l'on suppose dérivable au sens de Fomin et ayant les derivées partielles logarithmiques exponentiellement intégrables. On décrit une classe d'exemples a qui nos résultats s' appliquent et dont on peut aussi montrer l'unicité. Finalment on considère le cas où γ est la mesure invariante d'uneéquation de réaction-diffusion dont l'on prouve l'unicité des solutions. On utilise le fait que le gradient Dx est fermable dans L p (H, γ) et aussi une récente formule pour le commutateur DxPt − PtDx, Ptétant le sémigroupe de transitions qui corréspondà l'équation de réaction-diffusion considerée [DaDe14]. On souligne que dans ce cas Pt n'est pas nécessairement symétrique. Ce résultat d'unicité est une extension de celui obtenu dans [DaFlRoe14] ou γété la mesure invariante Gaussienne d'un processus de Ornstein-Uhlenbeck approprié.

Weak vorticity formulation of 2D Euler equations with white noise initial condition

“…Let µ be the law of white noise. Following [16], [18] and related literature, let us denote by FC 1 b,T the set of all functionals F : where ω ⊗ ω, H φ j , j = 1, ..., n, are the elements of L 2 (Ξ) given by Theorem 8. Hence D ω F (t, ω) , b (ω) is an element of C [0, T ] ; L 2 (Ξ) .…”

Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure

“…We may construct ρ N t and prove (12) also by the following procedure, closer to [7]. We study the transport equation in…”

“…for some ǫ > 0, which depends only on φ ∞ ; see Theorem 8 in Section 2 below. This exponential integrability is a key ingredient to extend, to the 2D Euler equations, the result of the authors [7] for abstract equations in Hilbert spaces (in that work the measure µ is not necessarily Gaussian, but the nonlinearity is bounded). Indeed, we aim to prove existence in the class of densities ρ t (ω) such that sup…”

“…Assume now that ρ 0 satisfies only the assumptions of the main theorem. By Corollory C.3 in [7], there exists a sequence ρ n 0 of bounded continuous functions (in fact bounded smooth cylinder functions) that converges to ρ 0 in L 1 (B, µ) and…”

“…Since ab ≤ e ǫa + ǫ −1 b log ǫ −1 b − 1 , if ρ t (ω) satisfies (7) and property ( 6) is proved, then…”