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2016 | Buch

Preliminaries, Notation, and Spaces of Functions Kapitel lesen Erstes Kapitel lesen

Mathematical Theory of Compressible Viscous Fluids

Analysis and Numerics

verfasst von: Eduard Feireisl, Trygve G. Karper, Milan Pokorný

Verlag: Springer International Publishing

Buchreihe : Advances in Mathematical Fluid Mechanics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers an essential introduction to the mathematical theory of compressible viscous fluids. The main goal is to present analytical methods from the perspective of their numerical applications. Accordingly, we introduce the principal theoretical tools needed to handle well-posedness of the underlying Navier-Stokes system, study the problems of sequential stability, and, lastly, construct solutions by means of an implicit numerical scheme. Offering a unique contribution – by exploring in detail the “synergy” of analytical and numerical methods – the book offers a valuable resource for graduate students in mathematics and researchers working in mathematical fluid mechanics.

Mathematical fluid mechanics concerns problems that are closely connected to real-world applications and is also an important part of the theory of partial differential equations and numerical analysis in general. This book highlights the fact that numerical and mathematical analysis are not two separate fields of mathematics. It will help graduate students and researchers to not only better understand problems in mathematical compressible fluid mechanics but also to learn something from the field of mathematical and numerical analysis and to see the connections between the two worlds. Potential readers should possess a good command of the basic tools of functional analysis and partial differential equations including the function spaces of Sobolev type.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries, Notation, and Spaces of Functions
Abstract
This chapter introduces notation as well as the basic mathematical tools used in the book such as the function spaces, embedding theorems, and elementary inequalities. We suppose the reader to be familiar with this material and will refer to it throughout the text without further specification.
Eduard Feireisl, Trygve G. Karper, Milan Pokorný

Mathematics of Compressible Fluid Flows

Frontmatter
Chapter 2. Mathematical Model
Abstract
As this book is focused on purely mathematical aspects of the theory of compressible viscous fluids, we omit a detailed derivation of the model in terms of classical continuum mechanics. The interested reader may consult the monographs of Batchelor [6], Lamb [65], or the more recent treatment by Gallavotti [47].
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 3. Weak Solutions
Abstract
A vast class of nonlinear evolutionary problems arising in mathematical fluid mechanics including the Navier–Stokes system (2.​7) and (2.​8) is not known to admit classical (differentiable, smooth) solutions for all choices of data and on an arbitrary time interval. The existence of classical solutions has been established under rather restrictive conditions, notably if
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 4. A Priori Bounds
Abstract
A priori bounds are natural constraints imposed on the set of (hypothetical) smooth solutions by the data as well as by the differential equations satisfied. A priori bounds determine the function spaces framework the (weak) solutions are looked for. By definition, they are formal, derived under the principal hypothesis of smoothness of all quantities in question. A priori bounds have their counter part in the stability estimates derived for the associated numerical scheme discussed in Part II. As we will see, the a priori bounds available for the Navier–Stokes system are rather poor and can be derived by means of elementary integration. Unfortunately, they are the best available for our problem although some substantial and mathematically rather delicate improvements have been obtained recently by Plotnikov and Weigant [81] in the simplified 2-D geometry, see also [82] for the stationary case.
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 5. Weak Formulation Revisited
Abstract
In Chap. 3, we have introduced the weak formulation of both the equation of continuity (3.​5) and the momentum balance (3.​11). On the other hand, we have seen in Chap. 4 that regular solutions of the Navier–Stokes system satisfy also the renormalized equation of continuity (4.​8), together with the total energy balance (4.​10). Under the general hypothesis considered in this book, the piece of information encoded in (4.​8) and (4.​10) cannot be obtained directly from the weak formulation (3.​5) and (3.​11). Accordingly, it seems convenient to include both (4.​8) and (4.​10) in the definition of weak solutions to the problem (2.​7), (2.​8), (2.​13)–(2.​15) as a kind of additional admissibility criteria.
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 6. Weak Sequential Stability
Abstract
The property of weak sequential stability plays a crucial role in the analysis of any nonlinear problem. It states that the solution set of a given problem is (weakly) precompact with respect to the topologies induced by the available a priori estimates. In our context, this property can be stated as follows:
Eduard Feireisl, Trygve G. Karper, Milan Pokorný

Existence of Weak Solutions via a Numerical Method

Frontmatter
Chapter 7. Numerical Method
Abstract
We show that the weak solutions to the Navier–Stokes system exist, globally in time, for any finite energy initial data. The proof will be constructive in the sense that the desired weak solution is obtained as a suitable limit of a numerical scheme. By a numerical scheme we mean a finite number of algebraic equations yielding an approximate solution of the problem. To this end, we use the method of time discretization in combination with a mixed finite-volume finite-element scheme to solve that resulting “stationary” problems. The scheme is implicit, the numerical approximation at any time level is obtained as a solution of a finite system of nonlinear algebraic equations resulting from the spatial discretization.
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 8. Stability of the Numerical Method
Abstract
Stability in numerical analysis means that the approximate solutions admit the same bounds as indicated by the a priori estimates for the original problem. The fact that the numerical solutions satisfy the energy inequality (7.​28) plays a crucial role.
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 9. Consistency
Abstract
We derive a consistency formulation of the numerical method (7.​14) and (7.​15). This amounts to rewriting the upwind and other spatial discretization operators in terms of conventional derivatives, extending validity of (7.​14) and (7.​15) to the class of smooth test functions, and identifying the resulting error terms. Consistency formulation then takes us back to the original problem for which the steps of Chap. 6 can be easily adapted.
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 10. Convergence
Abstract
We are finally ready to establish convergence of the family [ϱ h , u h ] of approximate (numerical) solutions, the existence of which is guaranteed by Proposition 1. We follow closely the arguments already used in the proof of weak sequential stability developed in Chap. 6 To begin observe that, in view of the uniform bounds (8.​17) and (8.​19),
Eduard Feireisl, Trygve G. Karper, Milan Pokorný

Existence Theory for General Pressure

Frontmatter
Chapter 11. Weak Solutions with Artificial Pressure
Abstract
The last part of this book is devoted to the proof of existence of weak solutions to the compressible Navier–Stokes system for a general pressure law p = p(ϱ).
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 12. Strong Convergence of the Approximate Densities
Abstract
In this chapter, we establish weak convergence of the densities for the perturbed Navier–Stokes system endowed with the artificial pressure (11.​2). Following the arguments used in Sect. 6.​3 we need the renormalized formulation of the continuity equation (5.​4). Unfortunately, the regularization technique of DiPerna and Lions [26] is no longer applicable as the limit density ϱ is not (known to be) square integrable. Instead, we exploit another piece of information hidden in the effective viscous flux identity (6.​1).
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Chapter 13. Concluding Remarks and Suggestions for Further Reading
Abstract
In the book, we deliberately focused on purely mathematical aspects omitting the physical background of the modeling of motion of compressible viscous fluids arising from classical continuum mechanics. The interested reader may consult the standard reference material by Batchelor [6], Lamb [65], Landau and Lifshitz [66], or the more recent treatment by Gallavotti [46, 47]. Mathematical aspects of the theory are accented in Chorin and Marsden [19], Truesdell [91, 92], or Truesdell and Rajagopal [93].
Eduard Feireisl, Trygve G. Karper, Milan Pokorný
Backmatter
Metadaten
Titel
Mathematical Theory of Compressible Viscous Fluids
verfasst von
Eduard Feireisl
Trygve G. Karper
Milan Pokorný
Copyright-Jahr
2016
Electronic ISBN
978-3-319-44835-0
Print ISBN
978-3-319-44834-3
DOI
https://doi.org/10.1007/978-3-319-44835-0