Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations

1. Front Matter

   Pages i-x

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2. The Problem Studied in This Book

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • Simon Markfelder

      Pages 3-12

   3. Hyperbolic Conservation Laws

      • Simon Markfelder

      Pages 13-25

   4. The Euler Equations as a Hyperbolic System of Conservation Laws

      • Simon Markfelder

      Pages 27-48

3. Convex Integration

   1. Front Matter

      Pages 49-49

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   2. Preparation for Applying Convex Integration to Compressible Euler

      • Simon Markfelder

      Pages 51-90

   3. Implementation of Convex Integration

      • Simon Markfelder

      Pages 91-143

4. Application to Particular Initial (Boundary) Value Problems

   1. Front Matter

      Pages 145-145

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   2. Infinitely Many Solutions of the Initial Boundary Value Problem for Barotropic Euler

      • Simon Markfelder

      Pages 147-152

   3. Riemann Initial Data in Two Space Dimensions for Isentropic Euler

      • Simon Markfelder

      Pages 153-183

   4. Riemann Initial Data in Two Space Dimensions for Full Euler

      • Simon Markfelder

      Pages 185-208

5. Back Matter

   Pages 209-242

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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.

• Admissible Weak Solutions
• Barotropic Euler Equations
• Barotropic Euler System
• Compressible Euler Equations
• Compressible Euler System
• Convex Integration
• Fluid Mechanics
• Ill-posedness
• Isentropic Euler Equations
• Well-posedness
• Isentropic Euler System
• Non-uniqueness
• Partial Differential Equations

• Department of Applied Math & Theoretical Physics (DAMTP), Centre for Math Sciences, University of Cambridge, Cambridge, UK

  Simon Markfelder

Simon Markfelder is currently a postdoctoral researcher at the University of Cambridge, United Kingdom. He completed his PhD at the University of Wuerzburg, Germany, in 2020 under the supervision of Christian Klingenberg. Simon Markfelder has published several papers in which he applies the convex integration technique to the compressible Euler equations.

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