Para Differential Calculus And Applications To The Cauchy Problem For Nonlinear Systems

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

Author : Guy Métivier
Publisher : Edizioni della Normale
ISBN 13 :
Total Pages : 170 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems by : Guy Métivier

Download or read book Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems written by Guy Métivier and published by Edizioni della Normale. This book was released on 2008-07-17 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Theory and Applications of Abstract Semilinear Cauchy Problems

Author : Pierre Magal
Publisher : Springer
ISBN 13 : 3030015068
Total Pages : 543 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Theory and Applications of Abstract Semilinear Cauchy Problems by : Pierre Magal

Download or read book Theory and Applications of Abstract Semilinear Cauchy Problems written by Pierre Magal and published by Springer. This book was released on 2018-11-21 with total page 543 pages. Available in PDF, EPUB and Kindle. Book excerpt: Several types of differential equations, such as functional differential equation, age-structured models, transport equations, reaction-diffusion equations, and partial differential equations with delay, can be formulated as abstract Cauchy problems with non-dense domain. This monograph provides a self-contained and comprehensive presentation of the fundamental theory of non-densely defined semilinear Cauchy problems and their applications. Starting from the classical Hille-Yosida theorem, semigroup method, and spectral theory, this monograph introduces the abstract Cauchy problems with non-dense domain, integrated semigroups, the existence of integrated solutions, positivity of solutions, Lipschitz perturbation, differentiability of solutions with respect to the state variable, and time differentiability of solutions. Combining the functional analysis method and bifurcation approach in dynamical systems, then the nonlinear dynamics such as the stability of equilibria, center manifold theory, Hopf bifurcation, and normal form theory are established for abstract Cauchy problems with non-dense domain. Finally applications to functional differential equations, age-structured models, and parabolic equations are presented. This monograph will be very valuable for graduate students and researchers in the fields of abstract Cauchy problems, infinite dimensional dynamical systems, and their applications in biological, chemical, medical, and physical problems.

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

Author : T. Alazard
Publisher : American Mathematical Soc.
ISBN 13 : 147043203X
Total Pages : 108 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations by : T. Alazard

Download or read book Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations written by T. Alazard and published by American Mathematical Soc.. This book was released on 2019-01-08 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Hyperbolic Partial Differential Equations and Geometric Optics

Author : Jeffrey Rauch
Publisher : American Mathematical Soc.
ISBN 13 : 0821872915
Total Pages : 386 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Hyperbolic Partial Differential Equations and Geometric Optics by : Jeffrey Rauch

Download or read book Hyperbolic Partial Differential Equations and Geometric Optics written by Jeffrey Rauch and published by American Mathematical Soc.. This book was released on 2012-05-01 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author : Ferruccio Colombini
Publisher : Springer
ISBN 13 : 3319520423
Total Pages : 308 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics by : Ferruccio Colombini

Download or read book Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics written by Ferruccio Colombini and published by Springer. This book was released on 2017-04-25 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Singular and Degenerate Cauchy Problems

Author :
Publisher : Academic Press
ISBN 13 : 008095636X
Total Pages : 343 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Singular and Degenerate Cauchy Problems by :

Download or read book Singular and Degenerate Cauchy Problems written by and published by Academic Press. This book was released on 1977-01-13 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering

Tools and Problems in Partial Differential Equations

Author : Thomas Alazard
Publisher : Springer Nature
ISBN 13 : 3030502848
Total Pages : 357 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Tools and Problems in Partial Differential Equations by : Thomas Alazard

Download or read book Tools and Problems in Partial Differential Equations written by Thomas Alazard and published by Springer Nature. This book was released on 2020-10-19 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Nonlinear Equations: Methods, Models and Applications

Author : Daniela Lupo
Publisher : Birkhäuser
ISBN 13 : 3034880871
Total Pages : 268 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Nonlinear Equations: Methods, Models and Applications by : Daniela Lupo

Download or read book Nonlinear Equations: Methods, Models and Applications written by Daniela Lupo and published by Birkhäuser. This book was released on 2012-12-06 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: A collection of research articles originating from the Workshop on Nonlinear Analysis and Applications held in Bergamo in July 2001. Classical topics of nonlinear analysis were considered, such as calculus of variations, variational inequalities, critical point theory and their use in various aspects of the study of elliptic differential equations and systems, equations of Hamilton-Jacobi, Schrödinger and Navier-Stokes, and free boundary problems. Moreover, various models were focused upon: travelling waves in supported beams and plates, vortex condensation in electroweak theory, information theory, non-geometrical optics, and Dirac-Fock models for heavy atoms.

Fourier Analysis

Author : Michael Ruzhansky
Publisher : Springer Science & Business Media
ISBN 13 : 3319025503
Total Pages : 416 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Fourier Analysis by : Michael Ruzhansky

Download or read book Fourier Analysis written by Michael Ruzhansky and published by Springer Science & Business Media. This book was released on 2014-01-18 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.”

Mathematics of Wave Phenomena

Author : Willy Dörfler
Publisher : Springer Nature
ISBN 13 : 3030471748
Total Pages : 330 pages
Book Rating : 4.0/5 (34 download)

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Book Synopsis Mathematics of Wave Phenomena by : Willy Dörfler

Download or read book Mathematics of Wave Phenomena written by Willy Dörfler and published by Springer Nature. This book was released on 2020-10-01 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wave phenomena are ubiquitous in nature. Their mathematical modeling, simulation and analysis lead to fascinating and challenging problems in both analysis and numerical mathematics. These challenges and their impact on significant applications have inspired major results and methods about wave-type equations in both fields of mathematics. The Conference on Mathematics of Wave Phenomena 2018 held in Karlsruhe, Germany, was devoted to these topics and attracted internationally renowned experts from a broad range of fields. These conference proceedings present new ideas, results, and techniques from this exciting research area.

The Cauchy Problem for Higher Order Abstract Differential Equations

Author : Ti-Jun Xiao
Publisher : Springer
ISBN 13 : 3540494790
Total Pages : 314 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis The Cauchy Problem for Higher Order Abstract Differential Equations by : Ti-Jun Xiao

Download or read book The Cauchy Problem for Higher Order Abstract Differential Equations written by Ti-Jun Xiao and published by Springer. This book was released on 2013-12-11 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main purpose of this book is to present the basic theory and some recent de velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.

Free Boundary Problems in Fluid Dynamics

Author : Albert Ai
Publisher : Springer Nature
ISBN 13 : 3031604520
Total Pages : 373 pages
Book Rating : 4.0/5 (316 download)

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Book Synopsis Free Boundary Problems in Fluid Dynamics by : Albert Ai

Download or read book Free Boundary Problems in Fluid Dynamics written by Albert Ai and published by Springer Nature. This book was released on with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Anomalies in Partial Differential Equations

Author : Massimo Cicognani
Publisher : Springer Nature
ISBN 13 : 3030613461
Total Pages : 469 pages
Book Rating : 4.0/5 (36 download)

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Book Synopsis Anomalies in Partial Differential Equations by : Massimo Cicognani

Download or read book Anomalies in Partial Differential Equations written by Massimo Cicognani and published by Springer Nature. This book was released on 2021-02-03 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: The contributions contained in the volume, written by leading experts in their respective fields, are expanded versions of talks given at the INDAM Workshop "Anomalies in Partial Differential Equations" held in September 2019 at the Istituto Nazionale di Alta Matematica, Dipartimento di Matematica "Guido Castelnuovo", Università di Roma "La Sapienza". The volume contains results for well-posedness and local solvability for linear models with low regular coefficients. Moreover, nonlinear dispersive models (damped waves, p-evolution models) are discussed from the point of view of critical exponents, blow-up phenomena or decay estimates for Sobolev solutions. Some contributions are devoted to models from applications as traffic flows, Einstein-Euler systems or stochastic PDEs as well. Finally, several contributions from Harmonic and Time-Frequency Analysis, in which the authors are interested in the action of localizing operators or the description of wave front sets, complete the volume.

The Cauchy Problem for Non-Lipschitz Semi-Linear Parabolic Partial Differential Equations

Author : J. C. Meyer
Publisher : Cambridge University Press
ISBN 13 : 1316301079
Total Pages : 177 pages
Book Rating : 4.3/5 (163 download)

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Book Synopsis The Cauchy Problem for Non-Lipschitz Semi-Linear Parabolic Partial Differential Equations by : J. C. Meyer

Download or read book The Cauchy Problem for Non-Lipschitz Semi-Linear Parabolic Partial Differential Equations written by J. C. Meyer and published by Cambridge University Press. This book was released on 2015-10-22 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reaction-diffusion theory is a topic which has developed rapidly over the last thirty years, particularly with regards to applications in chemistry and life sciences. Of particular importance is the analysis of semi-linear parabolic PDEs. This monograph provides a general approach to the study of semi-linear parabolic equations when the nonlinearity, while failing to be Lipschitz continuous, is Hölder and/or upper Lipschitz continuous, a scenario that is not well studied, despite occurring often in models. The text presents new existence, uniqueness and continuous dependence results, leading to global and uniformly global well-posedness results (in the sense of Hadamard). Extensions of classical maximum/minimum principles, comparison theorems and derivative (Schauder-type) estimates are developed and employed. Detailed specific applications are presented in the later stages of the monograph. Requiring only a solid background in real analysis, this book is suitable for researchers in all areas of study involving semi-linear parabolic PDEs.

The Cauchy Problem

Author : Hector O. Fattorini
Publisher : Cambridge University Press
ISBN 13 : 0521302382
Total Pages : 664 pages
Book Rating : 4.5/5 (213 download)

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Book Synopsis The Cauchy Problem by : Hector O. Fattorini

Download or read book The Cauchy Problem written by Hector O. Fattorini and published by Cambridge University Press. This book was released on 1983 with total page 664 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume deals with the Cauchy or initial value problem for linear differential equations. It treats in detail some of the applications of linear space methods to partial differential equations, especially the equations of mathematical physics such as the Maxwell, Schrödinger and Dirac equations. Background material presented in the first chapter makes the book accessible to mathematicians and physicists who are not specialists in this area as well as to graduate students.

The Cauchy Problem for Non-Lipschitz Semi-linear Parabolic Partial Differential Equations

Author : John Christopher Meyer
Publisher :
ISBN 13 : 9781316317778
Total Pages : 167 pages
Book Rating : 4.3/5 (177 download)

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Book Synopsis The Cauchy Problem for Non-Lipschitz Semi-linear Parabolic Partial Differential Equations by : John Christopher Meyer

Download or read book The Cauchy Problem for Non-Lipschitz Semi-linear Parabolic Partial Differential Equations written by John Christopher Meyer and published by . This book was released on 2015 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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

Author : Chao Wang
Publisher : American Mathematical Soc.
ISBN 13 : 1470446898
Total Pages : 119 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary by : Chao Wang

Download or read book Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary written by Chao Wang and published by American Mathematical Soc.. This book was released on 2021-07-21 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.