Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

Abstract

We prove uniqueness for continuity equations in Hilbert spaces H . The corresponding drift F is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different approximation procedure, based on a more regularizing OrnsteinUhlenbeck semigroup and consider Sobolev spaces of vector fields taking values in H rather than the Cameron-Martin space of the Gaussian measure. This leads to different conditions on the derivative of F, which are incompatible with previous work on the subject. Furthermore, we can drop the usual exponential integrability conditions on the Gaussian divergence of F, thus improving known uniqueness results in this respect.

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

“…In the case of Theorem 25 on ρ−white noise solutions, we are certainly far away from any uniqueness claim, even in law. Presumably one should try first to investigate uniqueness of ρ t , maybe with tools related to those of [7], [8], [18], [20], which already looks a formidable task.…”

“…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) .…”

Absolutely continuous solutions for continuity equations in Hilbert spaces

“…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.…”

“…This, in particular, covers the case studied in [DaFlRoe14], where only uniqueness of solutions to (1.3) was studied.…”

“…To our knowledge, earliest existence (and uniqueness) results for equation (1.3) concern the case where H is finite dimensional and the reference measure is the Lebesgue measure, see the seminal papers [DiLi89] and [Am04]. If H is infinite dimensional and γ is a Gaussian measure, problem (1.1) has been studied in [AmFi09], [FaLu10] and [DaFlRoe14]. In [KoRoe14] also non-Gaussian measures, γ, e.g.…”