Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems
Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems

Author: Guy Métivier

Publisher: Edizioni della Normale

Published: 2008-07-17

Total Pages: 170

ISBN-13:

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The main aim is to present at the level of beginners several modern tools of micro-local analysis which are useful for the mathematical study of nonlinear partial differential equations. The core of these notes is devoted to a presentation of the para-differential techniques, which combine a linearization procedure for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. These methods apply to many problems in nonlinear PDE’s such as elliptic equations, propagation of singularities, boundary value problems, shocks or boundary layers. However, in these introductory notes, we have chosen to illustrate the theory on two selected and relatively simple examples, which allow becoming familiar with the techniques. They concern the well posed-ness of the Cauchy problem for systems of nonlinear PDE's, firstly hyperbolic systems and secondly coupled systems of Schrödinger equations which arise in various models of wave propagation.

Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations

Author: T. Alazard

Publisher: American Mathematical Soc.

Published: 2019-01-08

Total Pages: 120

ISBN-13: 147043203X

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This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, the authors consider solutions such that the curvature of the initial free surface does not belong to L2. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. The authors first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.

Author:

Publisher: Springer Nature

Published:

Total Pages: 439

ISBN-13: 3031709098

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Hyperbolic Partial Differential Equations and Geometric Optics
Hyperbolic Partial Differential Equations and Geometric Optics

Author: Jeffrey Rauch

Publisher: American Mathematical Soc.

Published: 2012-05-01

Total Pages: 386

ISBN-13: 0821872915

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This book introduces graduate students and researchers in mathematics and the sciences to the multifaceted subject of the equations of hyperbolic type, which are used, in particular, to describe propagation of waves at finite speed. Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant interactions. Many fundamental results are presented for the first time in a textbook format. In addition to dense oscillations, these include the treatment of precise speed of propagation and the existence and stability questions for the three wave interaction equations. One of the strengths of this book is its careful motivation of ideas and proofs, showing how they evolve from related, simpler cases. This makes the book quite useful to both researchers and graduate students interested in hyperbolic partial differential equations. Numerous exercises encourage active participation of the reader. The author is a professor of mathematics at the University of Michigan. A recognized expert in partial differential equations, he has made important contributions to the transformation of three areas of hyperbolic partial differential equations: nonlinear microlocal analysis, the control of waves, and nonlinear geometric optics.

Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics
Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics

Author: Ferruccio Colombini

Publisher: Springer

Published: 2017-04-25

Total Pages: 313

ISBN-13: 3319520423

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The book collects the most relevant results from the INdAM Workshop "Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics" held in Rome, September 14-18, 2015. The contributions discuss recent major advances in the study of nonlinear hyperbolic systems, addressing general theoretical issues such as symmetrizability, singularities, low regularity or dispersive perturbations. It also investigates several physical phenomena where such systems are relevant, such as nonlinear optics, shock theory (stability, relaxation) and fluid mechanics (boundary layers, water waves, Euler equations, geophysical flows, etc.). It is a valuable resource for researchers in these fields.

Fourier Analysis
Fourier Analysis

Author: Michael Ruzhansky

Publisher: Springer Science & Business Media

Published: 2014-01-18

Total Pages: 416

ISBN-13: 3319025503

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This book is devoted to the broad field of Fourier analysis and its applications to several areas of mathematics, including problems in the theory of pseudo-differential operators, partial differential equations, and time-frequency analysis. It is based on lectures given at the international conference “Fourier Analysis and Pseudo-Differential Operators,” June 25–30, 2012, at Aalto University, Finland. This collection of 20 refereed articles is based on selected talks and presents the latest advances in the field. The conference was a satellite meeting of the 6th European Congress of Mathematics, which took place in Krakow in July 2012; it was also the 6th meeting in the series “Fourier Analysis and Partial Differential Equations.”

Evolution Equations of Hyperbolic and Schrödinger Type
Evolution Equations of Hyperbolic and Schrödinger Type

Author: Michael Ruzhansky

Publisher: Springer Science & Business Media

Published: 2012-08-04

Total Pages: 327

ISBN-13: 3034804547

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Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area. ​

Tools and Problems in Partial Differential Equations
Tools and Problems in Partial Differential Equations

Author: Thomas Alazard

Publisher: Springer Nature

Published: 2020-10-19

Total Pages: 357

ISBN-13: 3030502848

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This textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations. Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory. Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject.

Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary
Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary

Author: Chao Wang

Publisher: American Mathematical Soc.

Published: 2021-07-21

Total Pages: 119

ISBN-13: 1470446898

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In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to C 3 2 +ε. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.