On a rough perturbation of the Navier–Stokes system and its vorticity formulation

Abstract

We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of strong solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong-Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.

Solution properties of the incompressible Euler system with rough path advection

“…We showed that for arbitrarily given 0 ∈ ˚ 1 , there exists a time * = * ( , Z , | | 3,∞ ) and a solution ∈ 2 ([0, * ]; ˚ 2 ) ∩ ([0, * ]; ˚ 1 ). In fact, the solution was not constrained to be mean-free in [HLN21], but the proof goes through in a simpler manner.…”

“…It is worth noting that we impose the mean-free constraint because it simplifies our analysis. For details on how to avoid this assumption, we refer to [HLN21], which establishes the existence of a strong solution of the associated viscous version of equation (3.1).…”

“…−var is a continuous -rough path cocycle as defined, e.g., in [HLN21], then the 2 -system (RE) generates a continuous random dynamical system on ([0, max ); ˚ − ) for any > 0.…”

Stochastic Modelling of Small-Scale Perturbation

“…The asymptotic behaviour of (11) as → 0 can therefore be studied in a rigorous mathemathical framework as an example of Wong-Zakai approximation principle for stochastic PDEs. Starting from the seminal work of Wong and Zakai [27], a number of results have been obtained in this direction: we mention among others the works in [28][29][30][31][32][33] and, more recently, those in [34,35] based on rough path theory. The aforementioned results suggest, as a rule of thumb, to interpret the formal limit of v as a white-in-time noise in Stratonovich sense, that is, for every suitable process ϕ and some appropriate notion of convergence…”

Center Manifolds for Rough Partial Differential Equations

“…Rough path techniques provide a very natural framework to obtain RDS from SPDEs driven by general multiplicative noise since the usual problems with the nullsets do not appear in a pathwise approach. For instance, there are results regarding the existence of random dynamical systems generated by rough PDEs with transport [53,15], nonlinear multiplicative [49] and nonlinear conservative noise [28]. In this work we go beyond the of existence of RDS in the infinite-dimensional setting and establish a center manifold theorem (Theorem 6.8).…”