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Research in PDEs and related fields the 2019 Spring School, Sidi Bel Abbès, Algeria /

This volume presents an accessible overview of mathematical control theory and analysis of PDEs, providing young researchers a snapshot of these active and rapidly developing areas. The chapters are based on two mini-courses and additional talks given at the spring school "Trends in PDEs and Re...

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Other Authors: Ammari, Kaïs,, SpringerLink (Online service)
Format: eBook
Language: English
Published: Cham : Birkhäuser, [2022]
Physical Description: 1 online resource (vii, 186 pages) : illustrations (some color).
Series: Tutorials, schools, and workshops in the mathematical sciences.
Subjects:
Differential equations, Partial.
Differential equations, Partial -- Numerical solutions.
Control theory.
Table of Contents:
  • 880-01
  • 2.3 The Stokes Problem with Navier Boundary Condition
  • References
  • Survey on the Decay of the Local Energy for the Solutions of the Nonlinear Wave Equation
  • 1 Introduction and Preliminaries
  • 2 Scattering for the Subcritical and Critical Wave Equation
  • 2.1 The Subcritical Case
  • 2.1.1 Prisized Morawetz Estimate
  • 2.1.2 Global Time Strichartz Norms
  • 2.1.3 The Proof of Theorem 2.1
  • 2.2 The Critical Case
  • 2.2.1 Global Time Strichartz Norms
  • 2.2.2 The Proof of Theorem 2.1 in the Case p=5.
  • 3 Exponential Decay for the Local Energy of the Subcritical and Critical Wave Equation with Localized Semilinearity
  • 3.1 Nonlinear Lax-Phillips Theory
  • 3.2 Exponential Decay for the Local Energy of the Subcritical Wave Equation
  • 3.2.1 The Compactness of Z(T)
  • 3.2.2 Proof of Theorem 3.1
  • 3.3 Exponential Decay for the Local Energy of the Critical Wave Equation
  • 4 Polynomial Decay for the Local Energy of the Semilinear Wave Equation with Small Data
  • 4.1 Fundamental Lemmas
  • 4.2 Proof of Theorem 4.1: Existence and Decay of the Local Energy.
  • 5 Decay of the Local Energy for the Solutions of the Critical Klein-Gordon Equation
  • 5.1 Strichartz Norms Global in Time
  • 5.2 Exponential Decay of the Local Energy of Localized Linear Klein-Gordon Equation
  • 5.2.1 Semi-Group of Lax-Phillips Adapted to Localized Linear Klein-Gordon Equation
  • 5.2.2 Proof of Theorem 5.9
  • 5.3 Proof of Theorem 5.1
  • Appendix
  • References
  • A Spectral Numerical Method to Approximate the Boundary Controllability of the Wave Equation with Variable Coefficients
  • 1 Introduction
  • 2 Numerical Approximation of the Control Problem.
  • 3 Minimal L2-Weighted Controls
  • 4 Numerical Experiments
  • 5 Appendix
  • References
  • Aggregation Equation and Collapse to Singular Measure
  • 1 Introduction
  • 2 Graph Reformulation and Main Results
  • 3 Dini and Hölder Spaces
  • 4 Modified Curved Cauchy Operators
  • 5 Local Well-Posedness
  • 6 Global Well-Posedness
  • 6.1 Weak and Strong Damping Behavior of the Source Term
  • 6.2 Global a Priori Estimates
  • References
  • Geometric Control of Eigenfunctions of Schrödinger Operators
  • 1 Introduction
  • 2 The Geometric Control Condition.

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