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2022 | Buch

Sobolev Spaces and Elliptic Boundary Value Problems Kapitel lesen Erstes Kapitel lesen

Research in PDEs and Related Fields

The 2019 Spring School, Sidi Bel Abbès, Algeria

herausgegeben von: Kaïs Ammari

Verlag: Springer International Publishing

Buchreihe : Tutorials, Schools, and Workshops in the Mathematical Sciences

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

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Über dieses Buch

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 Related Fields” held at the University of Sidi Bel Abbès, Algeria from 8-10 April 2019. In addition to providing an in-depth summary of these two areas, chapters also highlight breakthroughs on more specific topics such as:

Sobolev spaces and elliptic boundary value problemsLocal energy solutions of the nonlinear wave equationGeometric control of eigenfunctions of Schrödinger operators

Research in PDEs and Related Fields will be a valuable resource to graduate students and more junior members of the research community interested in control theory and analysis of PDEs.

Inhaltsverzeichnis

Frontmatter
Sobolev Spaces and Elliptic Boundary Value Problems
Abstract
In this course, we are interested to the resolution of elliptical equations with various types of boundary conditions: Dirichlet, Neumann or Fourier-Robin, Navier or Navier type conditions. We will study the existence, the uniqueness, and the regularity of the solutions. This regularity will depend, as we will see, on data of the problem and, in particular, of the regularity of the domain. For the sake of clarity and simplification, we will examine two model problems, one involving the Laplacian and the other the Stokes operator.
Chérif Amrouche
Survey on the Decay of the Local Energy for the Solutions of the Nonlinear Wave Equation
Abstract
In this survey, we prove a Scattering theorem for the subcritical and critical wave equation outside convex obstacle. Then we deduce the exponential decay of local energy for localized semilinearity. The proof relies on generalized Strichartz estimates, and microlocal defect measures. We prove also that this decay is of polynomial type with general case of semilinearity but with small data.
Ahmed Bchatnia
A Spectral Numerical Method to Approximate the Boundary Controllability of the Wave Equation with Variable Coefficients
Abstract
We present a spectral numerical method to approximate the boundary control of the wave equation with a non-constant potential. The numerical implementation is described and some numerical experiments show the efficiency of the method.
Carlos Castro
Aggregation Equation and Collapse to Singular Measure
Abstract
We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments, and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.
Taoufik Hmidi, Dong Li
Geometric Control of Eigenfunctions of Schrödinger Operators
Abstract
We review the role of the Geometric Control Condition in establishing the observability property from an open set for solutions to the wave, Schrödinger, and eigenfunction equations. We show how to construct surfaces of revolution for which the observability property holds under strictly weaker conditions on the observation set than their counterparts for the wave and Schrödinger equations. We also introduce a class of Schrödinger operators on the two-dimensional sphere for which observability for eigenfunctions holds provided the observation region intersects only three fixed geodesics on the sphere, which only depend on the potential.
Fabricio Macià
Stability of a Graph of Strings with Local Kelvin–Voigt Damping
Abstract
We study the stability problem of a tree of elastic strings with local Kelvin–Voigt damping on some of the edges. Under appropriate conditions on the damping coefficients at the vertices, exponential/polynomial stability are proved. This is a new representation of Ammari et al. (Semigroup Forum 100:364–382, 2020), where we considered a tree. Then as indicated in paragraph four of Ammari et al. (Semigroup Forum 100:364–382, 2020), we obtain (under more generalized conditions on the damping coefficients) the same results.
Kaïs Ammari, Zhuangyi Liu, Farhat Shel
Metadaten
Titel
Research in PDEs and Related Fields
herausgegeben von
Kaïs Ammari
Copyright-Jahr
2022
Electronic ISBN
978-3-031-14268-0
Print ISBN
978-3-031-14267-3
DOI
https://doi.org/10.1007/978-3-031-14268-0

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