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Register a new userOn a rough perturbation of the Navier-Stokes system and its vorticity formulation
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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 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.
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This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space $holderspace{1}{alpha}$ which consists of differentiable functions whose first derivative is $alpha$ Holder continuous (see Section ref{sGexist} for the precise definition). Further, we show that as $N to infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(frac{1}{N})$ times its original energy as $t to infty$. Hence the limit $N to infty$ and $Tto infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t to infty$ explicitly.
We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for the Navier- Stokes system.
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.
We present a variational formulation for the Navier-Stokes-Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in cite{GBYo2016b} as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier-Stokes-Fourier system on Riemannian manifolds.
We consider point vortex systems on the two dimensional torus perturbed by environmental noise. It is shown that, under a suitable scaling of the noises, weak limit points of the empirical measures are solutions to the vorticity formulation of deterministic 2D Navier-Stokes equations.
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