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.
Author(s): Simon Markfelder
Series: Lecture Notes in Mathematics, 2294
Publisher: Springer
Year: 2021
Language: English
Pages: 252
City: Cham
Preface
Contents
Part I The Problem Studied in This Book
1 Introduction
1.1 The Euler Equations
1.2 Weak Solutions and Admissibility
1.3 Overview on Well-Posedness Results
1.4 Structure of This Book
2 Hyperbolic Conservation Laws
2.1 Formulation of a Conservation Law
2.2 Initial Boundary Value Problem
2.3 Hyperbolicity
2.4 Companion Laws and Entropies
2.5 Admissible Weak Solutions
3 The Euler Equations as a Hyperbolic Systemof Conservation Laws
3.1 Barotropic Euler System
3.1.1 Hyperbolicity
3.1.2 Entropies
3.1.3 Admissible Weak Solutions
3.2 Full Euler System
3.2.1 Hyperbolicity
3.2.2 Entropies
3.2.3 Admissible Weak Solutions
Part II Convex Integration
4 Preparation for Applying Convex Integrationto Compressible Euler
4.1 Outline and Preliminaries
4.1.1 Adjusting the Problem
4.1.2 Tartar's Framework
4.1.3 Plane Waves and the Wave Cone
4.1.4 Sketch of the Convex Integration Technique
4.2 -Convex Hulls
4.2.1 Definitions and Basic Facts
4.2.2 The HN-Condition and a Way to Define U
4.2.3 The -Convex Hull of Slices
4.2.4 The -Convex Hull if the Wave Cone is Complete
4.3 The Relaxed Set U Revisited
4.3.1 Definition of U
4.3.2 Computation of U
4.4 Operators
4.4.1 Statement of the Operators
4.4.2 Lemmas for the Proof of Proposition 4.4.1
4.4.3 Proof of Proposition 4.4.1
5 Implementation of Convex Integration
5.1 The Convex-Integration-Theorem
5.1.1 Statement of the Theorem
5.1.2 Functional Setup
5.1.3 The Functionals I0 and the Perturbation Property
5.1.4 Proof of the Convex-Integration-Theorem
5.2 Proof of the Perturbation Property
5.2.1 Lemmas for the Proof
5.2.2 Proof of Lemma 5.2.4
5.2.3 Proof of Lemma 5.2.1 Using Lemmas 5.2.2, 5.2.3and 5.2.4
5.2.4 Proof of the Perturbation Property Using Lemma 5.2.1
5.3 Convex Integration with Fixed Density
5.3.1 A Modified Version of the Convex-Integration-Theorem
5.3.2 Proof the Modified Perturbation Property
Part III Application to Particular Initial (Boundary) Value Problems
6 Infinitely Many Solutions of the Initial Boundary Value Problem for Barotropic Euler
6.1 A Simple Result on Weak Solutions
6.2 Possible Improvements to Obtain Admissible Weak Solutions
6.3 Further Possible Improvements
7 Riemann Initial Data in Two Space Dimensionsfor Isentropic Euler
7.1 One-Dimensional Self-Similar Solution
7.2 Summary of the Results on Non-/Uniqueness
7.3 Non-Uniqueness Proof if the Self-Similar Solution Consists of One Shock and One Rarefaction
7.3.1 Condition for Non-Uniqueness
7.3.2 The Corresponding System of Algebraic Equations and Inequalities
7.3.3 Simplification of the Algebraic System
7.3.4 Solution of the Algebraic System if the Rarefaction is ``Small''
7.3.5 Proof of Theorem 7.3.1 via an Auxiliary State
7.4 Sketches of the Non-Uniqueness Proofs for the Other Cases
7.4.1 Two Shocks
7.4.2 One Shock
7.4.3 A Contact Discontinuity and at Least One Shock
7.5 Other Results in the Context of the Riemann Problem
8 Riemann Initial Data in Two Space Dimensions for Full Euler
8.1 One-Dimensional Self-Similar Solution
8.2 Summary of the Results on Non-/Uniqueness
8.3 Non-Uniqueness Proof if the Self-Similar Solution Contains Two Shocks
8.3.1 Condition for Non-Uniqueness
8.3.2 The Corresponding System of Algebraic Equations and Inequalities
8.3.3 Solution of the Algebraic System
8.4 Sketches of the Non-Uniqueness Proofs for the Other Cases
8.4.1 One Shock and One Rarefaction
8.4.2 One Shock
8.5 Other Results in the Context of the Riemann Problem
A Notation and Lemmas
A.1 Sets
A.2 Vectors and Matrices
A.2.1 General Euclidean Spaces
A.2.2 The Physical Space and the Space-Time
A.2.3 Phase Space
A.3 Sequences
A.4 Functions
A.4.1 Basic Notions
A.4.2 Differential Operators
Functions of Time and Space
Functions of the State Vector
A.4.3 Function Spaces
A.4.4 Integrability Conditions
A.4.5 Boundary Integrals and the Divergence Theorem
A.4.6 Mollifiers
A.4.7 Periodic Functions
A.5 Convexity
A.5.1 Convex Sets and Convex Hulls
A.5.2 Convex Functions
A.6 Semi-Continuity
A.7 Weak- Convergence in L∞
A.8 Baire Category Theorem
Bibliography
Index
