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Item THE STOCHASTIC NAVIER STOKES EQUATIONS FOR HEAT CONDUCTING, COMPRESSIBLE FLUIDS(2016) Smith, Scott Andrew; Trivisa, Konstantina; Cerrai, Sandra; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation is devoted to the equations of motion governing the evolution of a fluid or gas at the macroscopic scale. The classical model is a PDE description known as the Navier-Stokes equations. The behavior of solutions is notoriously complex, leading many in the scientific community to describe fluid mechanics using a statistical language. In the physics literature, this is often done in an ad-hoc manner with limited precision about the sense in which the randomness enters the evolution equation. The stochastic PDE community has begun proposing precise models, where a random perturbation appears explicitly in the evolution equation. Although this has been an active area of study in recent years, the existing literature is almost entirely devoted to incompressible fluids. The purpose of this thesis is to take a step forward in addressing this statistical perspective in the setting of compressible fluids. In particular, we study the well posedness for the corresponding system of Stochastic Navier Stokes equations, satisfied by the density, velocity, and temperature. The evolution of the momentum involves a random forcing which is Brownian in time and colored in space. We allow for multiplicative noise, meaning that spatial correlations may depend locally on the fluid variables. Our main result is a proof of global existence of weak martingale solutions to the Cauchy problem set within a bounded domain, emanating from large initial datum. The proof involves a mix of deterministic and stochastic analysis tools. Fundamentally, the approach is based on weak compactness techniques from the deterministic theory combined with martingale methods. Four layers of approximate stochastic PDE's are built and analyzed. A careful study of the probability laws of our approximating sequences is required. We prove appropriate tightness results and appeal to a recent generalization of the Skorohod theorem. This ultimately allows us to deduce analogues of the weak compactness tools of Lions and Feireisl, appropriately interpreted in the stochastic setting.Item Mathematical Topics in Fluid-Particle Interaction(2014) Ballew, Joshua Thomas; Trivisa, Konstantina; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Models for particles interacting with compressible fluids are useful to several areas of science. This dissertation considers some of the mathematical issues of the Navier-Stokes-Smoluchowski and Euler-Smoluchowski models for compressible fluids. First, well-posedness for the NSS system is investigated. Among the results are the existence of weakly dissipative solutions obeying a relative entropy inequality. An approximating scheme using an artificial pressure and vanishing viscosity is employed to this end. The existence of these weakly dissipative solutions is used to show a weak-strong uniqueness result, using a Gronwall's argument on the relative entropy inequality. The existence of smooth solutions for finite time to the NSS system under certain compatibility conditions is shown using an iterative approximation. Next, two scaled regimes for the NSS system are considered. It is shown that for these low Mach number regimes, the solutions of the compressible system can be approximated by solutions of simpler models. In particular, the solutions to the model in a low stratification regime can be approximated by solutions to a model for incompressible flows with a Boussinesq relation. Solutions to the model in a strong stratification regime can be approximated by solutions to a model for anelastic flows. Much of the analysis for these limits relies on a Helmholtz free energy inequality, which bounds many of the quantities needed for the analysis. Lastly, the Euler-Smoluchowski model for inviscid, compressible fluids is considered. Finite-time existence of smooth solutions is shown using an iterative approximation and the results of Friedrichs and Majda for existence of smooth solutions for symmetric hyperbolic systems.Item Fast Solvers for Models of Fluid Flow with Spectral Elements(2008-09-02) Lott, Paul Aaron; Elman, Howard; Deane, Anil; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We introduce a preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion and the linearized Navier-Stokes equations. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this preconditioner in numerical simulations using a spectral element discretization. This work extends the use of Fast Diagonalization to steady convection-diffusion systems. We also extend the "least-squares commutator" preconditioner, originally developed for the finite element method, to a matrix-free spectral element framework. We show that these two advances, when used together, allow for efficient computation of steady-state solutions the the incompressible Navier-Stokes equations using high-order spectral element discretizations.Item Weakly Compressible Navier-Stokes Approximation of Gas Dynamics(2006-08-07) Jiang, Ning; Levermore, Charles David; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation addresses mathematical issues regarding weakly compressible approximations of gas dynamics that arise both in fluid dynamical and in kinetic settings. These approximations are derived in regimes in which (1) transport coefficients (viscosity and thermal conductivity) are small and (2) the gas is near an absolute equilibrium --- a spatially uniform, stationary state. When we consider regimes in which both the transport scales and $\mathrm{Re}$ vanish, we derive the {\em weakly compressible Stokes approximation} --- a {\em linear} system. When we consider regimes in which the transport scales vanish while $\mathrm{Re}$ maintains order unity, we derive the {\em weakly compressible Navier-Stokes approximation}---a {\em quadratic} system. Each of these weakly compressible approximations govern both the acoustic and the incompressible modes of the gas. In the fluid dynamical setting, our derivations begin with the fully compressible Navier-Stokes system. We show that the structure of the weakly compressible Navier-Stokes system ensures that it has global weak solutions, thereby extending the Leray theory for the incompressible Navier-Stokes system. Indeed, we show that this is the case in a general setting of hyperbolic-parabolic systems that possess an entropy under a structure condition (which is satisfied by the compressible Navier-Stokes system.) Moreover, we obtain a regularity result for the acoustic modes for the weakly compressible Navier-Stokes system. In the kinetic setting, our derivations begin with the Boltzmann equation. Our work extends earlier derivations of the incompressible Navier-Stokes system by the inclusion of the acoustic modes. We study the validity of these approximations in the setting of the DiPerna-Lions global solutions. Assuming that DiPerna-Lions solutions satisfy the local conservation law of energy, we use a relative entropy method to justify the weakly compressible Stokes approximation which unifies the Acoustic-Stokes limits result of Golse-Levermore, and to justify the weakly compressible Navier-Stokes approximation modulo further assumptions about passing to the limit in certain relative entropy dissipation terms. This last result extends the result of Golse-Levermore--Saint-Raymond for the incompressible Navier-Stokes system.