Stochastic Interacting Particle Systems and Nonlinear Partial Differential Equations from Fluid Mechanics

We derive stochastic particle approximations for two nonlinear partial differential equations from fluid mechanics: the porous medium equation and the three-dimensional Navier-Stokes equation. We associate interacting particle systems with these equations and obtain, when the number of particles tends to infinity, laws of large numbers for the empirical measures. In the first chapter we study a system of interacting diffusions and show that the empirical measure of the particle system tends to the solution of the porous medium equation when the number of particles tends to infinity. Moreover we prove propagation of chaos for this system: if initially the positions of the particles are independent and identically distributed, then they remain so - at least approximately - in the course of time. In the second chapter we consider a sequence of nonlinear stochastic differential equations and show that the distributions of the solutions converge to the solution of the viscous porous medium equation. The third chapter deals with a stochastic particle approximation for the three-dimensional Navier-Stokes equation. This equation is of a completely different type than the porous medium equation, so that it seems difficult to treat it with the methods of the first chapter. Nevertheless this is possible: we do not consider the Navier-Stokes equation directly, but instead the equation satisfied by the vorticity, and use the fact that the velocity can be recovered from the vorticity

Error estimation and adaptivity for incompressible, non–linear (hyper–)elasticity

A Galerkin finite element method is developed for non–linear, incompressible (hyper) elasticity, and a posteriori error estimates are derived for both linear functionals of the solution and linear functionals of the stress on a boundary where Dirichlet boundary conditions are applied. A second, higher order method for calculating a linear functional of the stress on a Dirichlet boundary is also presented together with an a posteriori error estimator for this approach. An implementation for a 2D model problem with known solution demonstrates the accuracy of the error estimators. Finally the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity

DESERT: Decision Support System for Evaluating River Basin Strategies

An integrated PC-based software package for decision support in water quality management on a river basin scale has been developed. The software incorporates a number of useful tools, including an easy-to-use data handling module with a dBase style database engine, simulation and calibration of hydraulics and water quality, display of computed data with the help of external spreadsheet software, and optimization based on dynamic programming algorithm. The main utility of the package is to provide useful and powerful instrument for water quality assessment and decision making in emission control, including selection of wastewater treatment alternatives, standard setting and enforcement at the river basin level. Two versions of the decision support software are presented, the current version and development of a follow-up program with extended features

Stochastic vortex method for forced three-dimensional Navier--Stokes equations and pathwise convergence rate

We develop a McKean-Vlasov interpretation of Navier-Stokes equations with external force field in the whole space, by associating with local mild $L^p$-solutions of the 3d-vortex equation a generalized nonlinear diffusion with random space-time birth that probabilistically describes creation of rotation in the fluid due to nonconservativeness of the force. We establish a local well-posedness result for this process and a stochastic representation formula for the vorticity in terms of a vector-weighted version of its law after its birth instant. Then we introduce a stochastic system of 3d vortices with mollified interaction and random space-time births, and prove the propagation of chaos property, with the nonlinear process as limit, at an explicit pathwise convergence rate. Convergence rates for stochastic approximation schemes of the velocity and the vorticity fields are also obtained. We thus extend and refine previous results on the probabilistic interpretation and stochastic approximation methods for the nonforced equation, generalizing also a recently introduced random space-time-birth particle method for the 2d-Navier-Stokes equation with force.Comment: Published in at http://dx.doi.org/10.1214/09-AAP672 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Soluciones débiles en mecánica de fluidos

The main aim of this work is to prove theoretical results on partial differential equations from fluid mechanics. Particularly, the theoretical development is destined to prove the existence of weak solutions of the Navier-Stokes equations in two and three dimensions. The Navier-Stokes equations is a classical topic in the study of the dynamics of incompressible viscous fluids. Those equations present basic and important open questions such as regularity and finite time singularity formation of the solutions. It is a current area of mathematical research of fundamental interest in particular due to its physical relevance and broad applicability. The first chapter introduces concepts of Functional Analysis that go beyond the scope of what is taught in the degree, and will be very useful in the development of this work. It establishes the main concepts and results related to weak convergence, needed to understand the concept of weak solution. The goal of the first section is to prove the weak compactness of bounded sets in a Hilbert space. Next, we take on the evolution problem of second-order parabolic equations, which has as a representative example the Heat equation. We use variational formulation to prove the existence of weak solutions. For that, we analyze the properties of the terms of the equation. On these properties we will prove some results of continuity and compactness, and we will finally apply Galerkin method. Due to the good properties of the equation, the results are proven in an arbitrary finite dimension and the uniqueness of the solution is proven as well. The techniques used for the study of the equation are repeated in the Navier-Stokes case. This first chapter also serves for acquiring familiarity with the method. The second chapter deals with the Navier-Stokes equation in the complete space of two and three dimensions with the same techniques as in the previous chapter. We will find greater difficulties due mainly to non-linearity. It begins by introducing the usual Hilbert spaces of fluid problems that incorporate incompressibility, and provides results that allow us to tackle the pressure of the equation, simplifying the problem. Next, we analyze in detail the non-linear term, finding a limitation in the dimension of the workspace. After introducing the variational formulation, compactness theorems which are necessary to treat the non-linear term are proven. Finally, the Galerkin method is applied again, and the existence of weak solutions in the cases of two and three dimensions is proved. The uniqueness in the two-dimensional case is also tested. This problem was originally studied by Jean Leray, who proved in 1934 the existence of weak solutions. For the three-dimensional case, it has recently been proven that there is no uniqueness of weak solutions.Universidad de Sevilla. Grado en Matemática

Dynamical density functional theory for molecular and colloidal fluids: a microscopic approach to fluid mechanics

In recent years, a number of dynamical density functional theories DDFTs have been developed for describing the dynamics of the one-body density of both colloidal and atomic fluids. In the colloidal case, the particles are assumed to have stochastic equations of motion and theories exist for both the case when the particle motion is overdamped and also in the regime where inertial effects are relevant. In this paper, we extend the theory and explore the connections between the microscopic DDFT and the equations of motion from continuum fluid mechanics. In particular, starting from the Kramers equation, which governs the dynamics of the phase space probability distribution function for the system, we show that one may obtain an approximate DDFT that is a generalization of the Euler equation. This DDFT is capable of describing the dynamics of the fluid density profile down to the scale of the individual particles. As with previous DDFTs, the dynamical equations require as input the Helmholtz free energy functional from equilibrium density functional theory DFT . For an equilibrium system, the theory predicts the same fluid one-body density profile as one would obtain from DFT. Making further approximations, we show that the theory may be used to obtain the mode coupling theory that is widely used for describing the transition from a liquid to a glassy state. © 2009 American Institute of Physics

A modified formal Lagrangian formulation for general differential equations

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.Comment: 18 page