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Convex Integration Applied To The Multi Dimensional Compressible Euler Equations
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Book Synopsis Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations by : Simon Markfelder
Download or read book Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations written by Simon Markfelder and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book applies the convex integration method to multi-dimensional compressible Euler equations in the barotropic case as well as the full system with temperature. The convex integration technique, originally developed in the context of differential inclusions, was applied in the groundbreaking work of De Lellis and Székelyhidi to the incompressible Euler equations, leading to infinitely many solutions. This theory was later refined to prove non-uniqueness of solutions of the compressible Euler system, too. These non-uniqueness results all use an ansatz which reduces the equations to a kind of incompressible system to which a slight modification of the incompressible theory can be applied. This book presents, for the first time, a generalization of the De Lellis-Székelyhidi approach to the setting of compressible Euler equations. The structure of this book is as follows: after providing an accessible introduction to the subject, including the essentials of hyperbolic conservation laws, the idea of convex integration in the compressible framework is developed. The main result proves that under a certain assumption there exist infinitely many solutions to an abstract initial boundary value problem for the Euler system. Next some applications of this theorem are discussed, in particular concerning the Riemann problem. Finally there is a survey of some related results. This self-contained book is suitable for both beginners in the field of hyperbolic conservation laws as well as for advanced readers who already know about convex integration in the incompressible framework.
Book Synopsis Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations by : Simon Markfelder
Download or read book Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations written by Simon Markfelder and published by Springer Nature. This book was released on 2021-10-20 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book applies the convex integration method to multi-dimensional compressible Euler equations in the barotropic case as well as the full system with temperature. The convex integration technique, originally developed in the context of differential inclusions, was applied in the groundbreaking work of De Lellis and Székelyhidi to the incompressible Euler equations, leading to infinitely many solutions. This theory was later refined to prove non-uniqueness of solutions of the compressible Euler system, too. These non-uniqueness results all use an ansatz which reduces the equations to a kind of incompressible system to which a slight modification of the incompressible theory can be applied. This book presents, for the first time, a generalization of the De Lellis–Székelyhidi approach to the setting of compressible Euler equations. The structure of this book is as follows: after providing an accessible introduction to the subject, including the essentials of hyperbolic conservation laws, the idea of convex integration in the compressible framework is developed. The main result proves that under a certain assumption there exist infinitely many solutions to an abstract initial boundary value problem for the Euler system. Next some applications of this theorem are discussed, in particular concerning the Riemann problem. Finally there is a survey of some related results. This self-contained book is suitable for both beginners in the field of hyperbolic conservation laws as well as for advanced readers who already know about convex integration in the incompressible framework.
Book Synopsis Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations by : Simon Markfelder
Download or read book Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations written by Simon Markfelder and published by Springer. This book was released on 2021-10-21 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book applies the convex integration method to multi-dimensional compressible Euler equations in the barotropic case as well as the full system with temperature. The convex integration technique, originally developed in the context of differential inclusions, was applied in the groundbreaking work of De Lellis and Székelyhidi to the incompressible Euler equations, leading to infinitely many solutions. This theory was later refined to prove non-uniqueness of solutions of the compressible Euler system, too. These non-uniqueness results all use an ansatz which reduces the equations to a kind of incompressible system to which a slight modification of the incompressible theory can be applied. This book presents, for the first time, a generalization of the De Lellis–Székelyhidi approach to the setting of compressible Euler equations. The structure of this book is as follows: after providing an accessible introduction to the subject, including the essentials of hyperbolic conservation laws, the idea of convex integration in the compressible framework is developed. The main result proves that under a certain assumption there exist infinitely many solutions to an abstract initial boundary value problem for the Euler system. Next some applications of this theorem are discussed, in particular concerning the Riemann problem. Finally there is a survey of some related results. This self-contained book is suitable for both beginners in the field of hyperbolic conservation laws as well as for advanced readers who already know about convex integration in the incompressible framework.
Book Synopsis Intermittent Convex Integration for the 3D Euler Equations by : Tristan Buckmaster
Download or read book Intermittent Convex Integration for the 3D Euler Equations written by Tristan Buckmaster and published by Princeton University Press. This book was released on 2023-07-11 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new threshold for the existence of weak solutions to the incompressible Euler equations To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L^2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates.
Book Synopsis On Positivity Preserving Finite Volume Schemes for Compressible Euler Equations by : Institute for Computer Applications in Science and Engineering
Download or read book On Positivity Preserving Finite Volume Schemes for Compressible Euler Equations written by Institute for Computer Applications in Science and Engineering and published by . This book was released on 1993 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Scientific and Technical Aerospace Reports by :
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1994 with total page 836 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners by : Paul J. Godin
Download or read book The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners written by Paul J. Godin and published by . This book was released on 1900 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "We study 2D compressible Euler flows in bounded impermeable domains whose boundary is smooth except for corners. We assume that the angles of the corners are small enough. Then we obtain local (in time) existence of solutions which keep the L2 Sobolev regularity of their Cauchy data, provided the external forces are sufficiently regular and suitable compatibility conditions are satisfied. Such a result is well known when there is no corner. Our proof relies on the study of associated linear problems. We also show that our results are rather sharp: we construct counterexamples in which the smallness condition on the angles is not fulfilled and which display a loss of L2 Sobolev regularity with respect to the Cauchy data and the external forces"--
Book Synopsis Multidimensional Singular Integrals and Integral Equations by : Solomon Grigorʹevich Mikhlin
Download or read book Multidimensional Singular Integrals and Integral Equations written by Solomon Grigorʹevich Mikhlin and published by Pergamon. This book was released on 1965 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multidimensional Singular Integrals and Integral Equations presents the results of the theory of multidimensional singular integrals and of equations containing such integrals. Emphasis is on singular integrals taken over Euclidean space or in the closed manifold of Liapounov and equations containing such integrals.
Book Synopsis Multi-dimensional Fluctuation Splitting Schemes for the Euler Equations on Unstructured Grids by : Lisa Marie Mesaros
Download or read book Multi-dimensional Fluctuation Splitting Schemes for the Euler Equations on Unstructured Grids written by Lisa Marie Mesaros and published by . This book was released on 1995 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Methods for Conservation Laws by : LEVEQUE
Download or read book Numerical Methods for Conservation Laws written by LEVEQUE and published by Birkhäuser. This book was released on 2013-11-11 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the following spring. The overall emphasis is on studying the mathematical tools that are essential in de veloping, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, particularly for problems involving shock waves. A reasonable un derstanding of the mathematical structure of these equations and their solutions is first required, and Part I of these notes deals with this theory. Part II deals more directly with numerical methods, again with the emphasis on general tools that are of broad use. I have stressed the underlying ideas used in various classes of methods rather than present ing the most sophisticated methods in great detail. My aim was to provide a sufficient background that students could then approach the current research literature with the necessary tools and understanding. vVithout the wonders of TeX and LaTeX, these notes would never have been put together. The professional-looking results perhaps obscure the fact that these are indeed lecture notes. Some sections have been reworked several times by now, but others are still preliminary. I can only hope that the errors are not too blatant. Moreover, the breadth and depth of coverage was limited by the length of these courses, and some parts are rather sketchy.
Book Synopsis Multidimensional Singular Integrals and Integral Equations by : Solomon Grigorʹevich Mikhlin
Download or read book Multidimensional Singular Integrals and Integral Equations written by Solomon Grigorʹevich Mikhlin and published by . This book was released on 1965 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Convex Integration Theory by : David Spring
Download or read book Convex Integration Theory written by David Spring and published by Springer Science & Business Media. This book was released on 2010-12-02 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: §1. Historical Remarks Convex Integration theory, ?rst introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov’s thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods.
Book Synopsis Handbook of Mathematical Fluid Dynamics by : S. Friedlander
Download or read book Handbook of Mathematical Fluid Dynamics written by S. Friedlander and published by Gulf Professional Publishing. This book was released on 2003-03-27 with total page 627 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Handbook of Mathematical Fluid Dynamics is a compendium of essays that provides a survey of the major topics in the subject. Each article traces developments, surveys the results of the past decade, discusses the current state of knowledge and presents major future directions and open problems. Extensive bibliographic material is provided. The book is intended to be useful both to experts in the field and to mathematicians and other scientists who wish to learn about or begin research in mathematical fluid dynamics. The Handbook illuminates an exciting subject that involves rigorous mathematical theory applied to an important physical problem, namely the motion of fluids.
Book Synopsis Handbook of Differential Equations: Evolutionary Equations by : C.M. Dafermos
Download or read book Handbook of Differential Equations: Evolutionary Equations written by C.M. Dafermos and published by Gulf Professional Publishing. This book was released on 2005-11-30 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains several introductory texts concerning the main directions in the theory of evolutionary partial differential equations. The main objective is to present clear, rigorous, and in depth surveys on the most important aspects of the present theory.
Book Synopsis Computing Information Directory by : Darlene Myers Hildebrandt
Download or read book Computing Information Directory written by Darlene Myers Hildebrandt and published by . This book was released on 1995 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Transport Processes at Fluidic Interfaces by : Dieter Bothe
Download or read book Transport Processes at Fluidic Interfaces written by Dieter Bothe and published by Birkhäuser. This book was released on 2017-07-13 with total page 679 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are several physico-chemical processes that determine the behavior of multiphase fluid systems – e.g., the fluid dynamics in the different phases and the dynamics of the interface(s), mass transport between the fluids, adsorption effects at the interface, and transport of surfactants on the interface – and result in heterogeneous interface properties. In general, these processes are strongly coupled and local properties of the interface play a crucial role. A thorough understanding of the behavior of such complex flow problems must be based on physically sound mathematical models, which especially account for the local processes at the interface. This book presents recent findings on the rigorous derivation and mathematical analysis of such models and on the development of numerical methods for direct numerical simulations. Validation results are based on specifically designed experiments using high-resolution experimental techniques. A special feature of this book is its focus on an interdisciplinary research approach combining Applied Analysis, Numerical Mathematics, Interface Physics and Chemistry, as well as relevant research areas in the Engineering Sciences. The contributions originated from the joint interdisciplinary research projects in the DFG Priority Programme SPP 1506 “Transport Processes at Fluidic Interfaces.”
Download or read book Physics Briefs written by and published by . This book was released on 1992 with total page 1040 pages. Available in PDF, EPUB and Kindle. Book excerpt:
