Convex integration applied to the multi-dimensional compressible Euler equations / / Simon Markfelder

• Record Nr.: UNISA996466386703316
• Autore: Markfelder Simon
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-83785-8
• Descrizione fisica: 1 online resource (244 pages)
• Collana: Lecture Notes in Mathematics ; ; v.2294
• Classificazione: 35Q31; 76N10; 35L65; 35L45; 35L50
• Disciplina: 515.35
• Soggetti: Differential equations; Physics; Global analysis (Mathematics); Equacions de Lagrange; Funcions convexes; Integració numèrica; Problemes de contorn; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.