Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.     

Diffusion-Approximations for Navier-Stokes Equation in Space-Time Gaussian Velocity Field

Abstract

By approximation methods, the existence of solutions for the Navier-Stokes equation with rapidly oscillating drift term for dimention 2 or 3 is proved. Also by the martingale method, stochastic Navier-Stokes equation is considered.     

Conditions on the Pressure for Vanishing Velocity in the Incompressible Fluid Flows in ℝ N

In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in ? ^N .     

Normal Feedback Stabilization of Periodic Flows in a Three-Dimensional Channel

We present an extension from two dimensions to three dimensions of a boundary control law, which stabilizes the parabolic profile of an infinite channel flow. The controller acts on the normal component of the velocity only. The stability is achieved without any a priori condition on the viscosity coefficient, that is on Reynolds number.     

The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels

Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). In this Letter, similarity transformation for the Navier-Stokes equation for 2-dimensional steady slip flow in microchannels is given. We provide an analytical solution for the slip flow using a powerful, easy-to-use analytic technique for non-linear problems, that is, the homotopy analysis method (HAM). The analytical solution is presented in the form of an infinite series. The effects of the Knudsen number (Kn) is discussed on the velocity profiles. It is found that the results are in excellent agreement with the existing results in the literature for the case of laminar developed flow.     

Pressure-driven transient flows of Newtonian fluids through microtubes with slip boundary

Recent advances in microscale experiments and molecular simulations confirm that slip of fluid on solid surface occurs at small scale, and thus the traditional no-slip boundary condition in fluid mechanics cannot be applied to flow in micrometer and nanometer scale tubes and channels. On the other hand, there is an urgent need to understand fluid flow in micrometer scale due to the emergence of biochemical lab-on-the-chip system and micro-electromechanical system fabrication technologies. In this paper, we study the pressure driven transient flow of an incompressible Newtonian fluid in microtubes with a Navier slip boundary condition. An exact solution is derived and is shown to include some existing known results as special cases. Through analysis of the derived solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the boundary slip does not alter the interior material deformation and stress field; while, for pressure fields with a wave form pressure gradient, the boundary slip causes the change of interior material deformation and consequently the velocity profile and stress field. We also derive asymptotic expressions for the exact solution through which a parameter is identified to dominate the behaviour of the flow driven by the wave form pressure gradient, and an explicit formulae for the critical slip parameter leading to the maximum transient flow rate is established.     

A continuum model of gas flows with localized density variations

We discuss the kinetic representation of gases and the derivation of macroscopic equations governing the thermomechanical behavior of a dilute gas viewed at the macroscopic level as a continuous medium. We introduce an approach to kinetic theory where spatial distributions of the molecules are incorporated through a mean-free-volume argument. The new kinetic equation derived contains an extra term involving the evolution of this volume, which we attribute to changes in the thermodynamic properties of the medium. Our kinetic equation leads to a macroscopic set of continuum equations in which the gradients of thermodynamic properties, in particular density gradients, impact on diffusive fluxes. New transport terms bearing both convective and diffusive natures arise and are interpreted as purely macroscopic expansion or compression. Our new model is useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluid flows.     

Gibbsian dynamics and ergodicity of magnetohydrodynamics and related systems forced by random noise

The magnetohydrodynamics system consists of the Navier-Stokes equations from fluid mechanics, coupled with the Maxwell’s equations from electromagnetism through multiples of non-linear terms involving derivatives. Following the approach of [1 Weinan, E., Mattingly, J. C., Sinai, Y. (2001). Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun. Math. Phys. 224:83–106. DOI:10.1007/s002201224083.[Crossref], [Web of Science ®] , [Google Scholar]], we prove the existence of a unique invariant measure in case the forcing terms consist of the cylindrical Wiener processes with only low modes. Its proof requires taking advantage of the structure of the non-linear terms carefully and is extended to various other related models such as the magnetohydrodynamics-Boussinesq system from fluid mechanics in atmosphere and oceans, as well as the magneto-micropolar fluid system from the theory of microfluids.