Asymptotic behaviour of a rapidly rotating fluid with random stationary surface stress

The goal of this paper is to describe in mathematical terms the effect on the ocean circulation of a random stationary wind stress at the surface of the ocean. In order to avoid singular behaviour, non-resonance hypotheses are introduced, which ensure that the time frequencies of the wind-stress are different from that of the Earth rotation. We prove a convergence result for a three-dimensional Navier-Stokes-Coriolis system in a bounded domain, in the asymptotic of fast rotation and vanishing vertical viscosity, and we exhibit some random and stationary boundary layer profiles. At last, an average equation is derived for the limit system in the case of the non-resonant torus.Comment: 45 page

A transmission problem across a fractal self-similar interface

We consider a transmission problem in which the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains occurs only at the fractal component of the interface between the interior and exterior domains. We also consider the sequence of the transmission problems in which the interior domain is obtained by stopping the self-similar construction after a finite number of steps; the transmission condition is then posed on a prefractal approximation of the fractal interface. We prove the convergence in the sense of Mosco of the energy forms associated with these problems to the energy form of the limit problem. In particular, this implies the convergence of the solutions of the approximated problems to the solution of the problem with fractal interface. The proof relies in particular on an extension property. Emphasis is put on the geometry of the ramified domain. The convergence result is obtained when the fractal interface has no self-contact, and in a particular geometry with self-contacts, for which an extension result is proved

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain $\Omega$ of R N , N=2,3, surrounded by a thin layer $\Sigma \epsilon$, along a part $\Gamma$2 of its boundary $\partial \Omega$, we consider a Navier-Stokes flow in $\Omega \cup \partial \Omega \cup \Sigma \epsilon$ with Reynolds' number of order 1/$\epsilon$ in $\Sigma \epsilon$. Using $\Gamma$-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface $\Gamma$2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context

A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games

Conservation laws arising in the study of forward-forward Mean-Field Games

We consider forward-forward Mean Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models

Fast and reliable pricing of American options with local volatility

We present globally convergent multigrid methods for the nonsymmetric obstacle problems as arising from the discretization of Black—Scholes models of American options with local volatilities and discrete data. No tuning or regularization parameters occur. Our approach relies on symmetrization by transformation and data recovery by superconvergence

Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable

The Navier wall law at a boundary with random roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size \eps \ll 1. In a parent paper, we derived a homogenized boundary condition of Navier type as \eps \to 0. We show here that for a large class of boundaries, this Navier condition provides a O(\eps^{3/2} |\ln \eps|^{1/2}) approximation in $L^2$, instead of O(\eps^{3/2}) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables

A global method for coupling transport with chemistry in heterogeneous porous media

Modeling reactive transport in porous media, using a local chemical equilibrium assumption, leads to a system of advection-diffusion PDE's coupled with algebraic equations. When solving this coupled system, the algebraic equations have to be solved at each grid point for each chemical species and at each time step. This leads to a coupled non-linear system. In this paper a global solution approach that enables to keep the software codes for transport and chemistry distinct is proposed. The method applies the Newton-Krylov framework to the formulation for reactive transport used in operator splitting. The method is formulated in terms of total mobile and total fixed concentrations and uses the chemical solver as a black box, as it only requires that on be able to solve chemical equilibrium problems (and compute derivatives), without having to know the solution method. An additional advantage of the Newton-Krylov method is that the Jacobian is only needed as an operator in a Jacobian matrix times vector product. The proposed method is tested on the MoMaS reactive transport benchmark.Comment: Computational Geosciences (2009) http://www.springerlink.com/content/933p55085742m203/?p=db14bb8c399b49979ba8389a3cae1b0f&pi=1