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Register a new userGeneralized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes
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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.
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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 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.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $alpha$-stable processes, which help verify the solvability of the original HJB equation.
In this paper, we are concerned with the local-in-time well-posedness of a fluid-kinetic model in which the BGK model with density dependent collision frequency is coupled with the inhomogeneous Navier-Stokes equation through drag forces. To the best knowledge of authors, this is the first result on the existence of local-in-time smooth solution for particle-fluid model with nonlinear inter-particle operator for which the existence of time can be prolonged as the size of initial data gets smaller.
This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion begin{eqnarray} left{begin{array}{lll} n_t+ucdot abla n= ablacdot(| abla n|^{p-2} abla n)- ablacdot(nchi(c) abla c),& xinOmega, t>0, c_t+ucdot abla c=Delta c-nf(c),& xinOmega, t>0, u_t+(ucdot abla) u=Delta u+ abla P+n ablaPhi,& xinOmega, t>0, ablacdot u=0,& xinOmega, t>0 end{array}right. end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Omegasubset mathbb{R}^3$ with smooth boundary. Here, $Phiin W^{1,infty}(Omega)$, $0<chiin C^2([0,infty))$ and $0leq fin C^1([0,infty))$ with $f(0)=0$. It is proved that if $p>frac{32}{15}$ and under appropriate structural assumptions on $f$ and $chi$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.
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