On the Navier–Stokes equation perturbed by rough transport noise

Abstract

We consider the Navier-Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space-time dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier-Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise. MSC Classification Numbers: 60H15, 76D05, 47J30, 60H05, 35A15.

Generalized Burgers equation with rough transport noise

“…The above corresponds to the remainder term in the theory of controlled rough paths, and we see that u ♭ ∈ C α * 2 ([0, T ]; H −1 ). Moreover, we can write 20) so that also u ♭ ∈ C 2α 2 ([0, T ]; H −3 ). In fact, we can interpolate between these two spaces as follows.…”

“…It should be mentioned that the work [20] consider the Navier-Stokes equation in the same framework as the present paper. However, there it is assumed that the vector fields β j are energy preserving, i.e.…”

An Itô Formula for rough partial differential equations and some applications

“…What plays here the role of the driving rough path for controlled differential equations is the pair B = (B 1 , B 2 ). It is called an unbounded rough driver (URD), and was first considered by Bailleul and Gubinelli [4] (see also [19,38,39]). In the present work, we chose to restrict our attention to a subclass of URDs that are given by differential operators.…”

“…In particular, they are simultaneously symmetric, closed and renormalizable in the sense of [4, definitions 5.3, 5.4 & 5.7]. In contrast with the previous works [19,38,39], we will be able to consider these objects "as such", in the sense that we will not refer to any (geometric) finite-dimensional rough path. This observation, which can be seen as one of our main contributions, allows us to gain generality in the statements and, hopefully, to improve the clarity of the presentation.…”

“…Concerning the last one, it relies on the following Aubin-Lions-type compactness result. The proof follows exactly the same steps as[39, Lemma A.2 & Lemma A.3], and is therefore omitted.…”