Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables

Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables
Title Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables PDF eBook
Author A. Majda
Publisher Springer Science & Business Media
Pages 167
Release 2012-12-06
Genre Science
ISBN 1461211166

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Conservation laws arise from the modeling of physical processes through the following three steps: 1) The appropriate physical balance laws are derived for m-phy- t cal quantities, ul""'~ with u = (ul' ... ,u ) and u(x,t) defined m for x = (xl""'~) E RN (N = 1,2, or 3), t > 0 and with the values m u(x,t) lying in an open subset, G, of R , the state space. The state space G arises because physical quantities such as the density or total energy should always be positive; thus the values of u are often con strained to an open set G. 2) The flux functions appearing in these balance laws are idealized through prescribed nonlinear functions, F.(u), mapping G into J j = 1, ..• ,N while source terms are defined by S(u,x,t) with S a given smooth function of these arguments with values in Rm. In parti- lar, the detailed microscopic effects of diffusion and dissipation are ignored. 3) A generalized version of the principle of virtual work is applied (see Antman [1]). The formal result of applying the three steps (1)-(3) is that the m physical quantities u define a weak solution of an m x m system of conservation laws, o I + N(Wt'u + r W ·F.(u) + W·S(u,x,t))dxdt (1.1) R xR j=l Xj J for all W E C~(RN x R+), W(x,t) E Rm.

Numerical Analysis of Compressible Fluid Flows

Numerical Analysis of Compressible Fluid Flows
Title Numerical Analysis of Compressible Fluid Flows PDF eBook
Author Eduard Feireisl
Publisher Springer Nature
Pages 530
Release 2022-01-01
Genre Mathematics
ISBN 3030737888

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This book is devoted to the numerical analysis of compressible fluids in the spirit of the celebrated Lax equivalence theorem. The text is aimed at graduate students in mathematics and fluid dynamics, researchers in applied mathematics, numerical analysis and scientific computing, and engineers and physicists. The book contains original theoretical material based on a new approach to generalized solutions (dissipative or measure-valued solutions). The concept of a weak-strong uniqueness principle in the class of generalized solutions is used to prove the convergence of various numerical methods. The problem of oscillatory solutions is solved by an original adaptation of the method of K-convergence. An effective method of computing the Young measures is presented. Theoretical results are illustrated by a series of numerical experiments. Applications of these concepts are to be expected in other problems of fluid mechanics and related fields.

Analysis of Systems of Conservation Laws

Analysis of Systems of Conservation Laws
Title Analysis of Systems of Conservation Laws PDF eBook
Author Heinrich Freistuhler
Publisher CRC Press
Pages 276
Release 1998-12-30
Genre Mathematics
ISBN 9780849306440

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Systems of partial differential equations reflecting conservation laws hold significant relevance to a variety of theoretical and practical applications, including compressible fluid flow, electromagnetism, elasticity theory, and other areas of continuum mechanics. This field of nonlinear analysis is currently experiencing a marked increase in successful research activity. The EU-TMR network "Hyperbolic Systems of Conservation Laws held a summer program offering short courses on the Analysis of Systems of Conservation Laws. This book contains five of the self-contained short courses presented during this program by experts of international reputation. These courses, which address solutions to hyperbolic systems by the front tracking method, non-strictly hyperbolic conservation laws, hyperbolic-elliptic coupled systems, hyperbolic relaxation problems, the stability of nonlinear waves in viscous media and numerics, and more, represent the state of the art of most central aspects of the field.

Nonlinear Conservation Laws and Applications

Nonlinear Conservation Laws and Applications
Title Nonlinear Conservation Laws and Applications PDF eBook
Author Alberto Bressan
Publisher Springer Science & Business Media
Pages 487
Release 2011-04-19
Genre Mathematics
ISBN 1441995544

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This volume contains the proceedings of the Summer Program on Nonlinear Conservation Laws and Applications held at the IMA on July 13--31, 2009. Hyperbolic conservation laws is a classical subject, which has experienced vigorous growth in recent years. The present collection provides a timely survey of the state of the art in this exciting field, and a comprehensive outlook on open problems. Contributions of more theoretical nature cover the following topics: global existence and uniqueness theory of one-dimensional systems, multidimensional conservation laws in several space variables and approximations of their solutions, mathematical analysis of fluid motion, stability and dynamics of viscous shock waves, singular limits for viscous systems, basic principles in the modeling of turbulent mixing, transonic flows past an obstacle and a fluid dynamic approach for isometric embedding in geometry, models of nonlinear elasticity, the Monge problem, and transport equations with rough coefficients. In addition, there are a number of papers devoted to applications. These include: models of blood flow, self-gravitating compressible fluids, granular flow, charge transport in fluids, and the modeling and control of traffic flow on networks.

Stochastically Forced Compressible Fluid Flows

Stochastically Forced Compressible Fluid Flows
Title Stochastically Forced Compressible Fluid Flows PDF eBook
Author Dominic Breit
Publisher Walter de Gruyter GmbH & Co KG
Pages 344
Release 2018-01-22
Genre Mathematics
ISBN 3110492555

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This book contains a first systematic study of compressible fluid flows subject to stochastic forcing. The bulk is the existence of dissipative martingale solutions to the stochastic compressible Navier-Stokes equations. These solutions are weak in the probabilistic sense as well as in the analytical sense. Moreover, the evolution of the energy can be controlled in terms of the initial energy. We analyze the behavior of solutions in short-time (where unique smooth solutions exists) as well as in the long term (existence of stationary solutions). Finally, we investigate the asymptotics with respect to several parameters of the model based on the energy inequality. Contents Part I: Preliminary results Elements of functional analysis Elements of stochastic analysis Part II: Existence theory Modeling fluid motion subject to random effects Global existence Local well-posedness Relative energy inequality and weak–strong uniqueness Part III: Applications Stationary solutions Singular limits

Nonlinear Conservation Laws, Fluid Systems and Related Topics

Nonlinear Conservation Laws, Fluid Systems and Related Topics
Title Nonlinear Conservation Laws, Fluid Systems and Related Topics PDF eBook
Author Gui-Qiang Chen
Publisher World Scientific
Pages 401
Release 2009
Genre Mathematics
ISBN 9814273287

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This book is a collection of lecture notes on Nonlinear Conservation Laws, Fluid Systems and Related Topics delivered at the 2007 Shanghai Mathematics Summer School held at Fudan University, China, by world''s leading experts in the field. The volume comprises five chapters that cover a range of topics from mathematical theory and numerical approximation of both incompressible and compressible fluid flows, kinetic theory and conservation laws, to statistical theories for fluid systems. Researchers and graduate students who want to work in this field will benefit from this essential reference as each chapter leads readers from the basics to the frontiers of the current research in these areas.

Handbook of Mathematical Fluid Dynamics

Handbook of Mathematical Fluid Dynamics
Title Handbook of Mathematical Fluid Dynamics PDF eBook
Author S. Friedlander
Publisher Elsevier
Pages 725
Release 2007-05-16
Genre Science
ISBN 0080478301

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This is the fourth volume in a series of survey articles covering many aspects of mathematical fluid dynamics, a vital source of open mathematical problems and exciting physics.