Nonlinear Evolution Equations
Nonlinear Evolution Equations

Author: Nina B. Maslova

Publisher: World Scientific

Published: 1993

Total Pages: 210

ISBN-13: 9789810211622

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The book is devoted to the questions of the long-time behavior of solutions for evolution equations, connected with kinetic models in statistical physics. There is a wide variety of problems where such models are used to obtain reasonable physical as well as numerical results (Fluid Mechanics, Gas Dynamics, Plasma Physics, Nuclear Physics, Turbulence Theory etc.). The classical examples provide the nonlinear Boltzmann equation. Investigation of the long-time behavior of the solutions for the Boltzmann equation gives an approach to the nonlinear fluid dynamic equations. From the viewpoint of dynamical systems, the fluid dynamic equations arise in the theory as a tool to describe an attractor of the kinetic equation.

Nonlinear Evolution Equations: Kinetic Approach
Nonlinear Evolution Equations: Kinetic Approach

Author: Niva B Maslova

Publisher: World Scientific

Published: 1993-03-10

Total Pages: 216

ISBN-13: 9814505161

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The book is devoted to the questions of the long-time behavior of solutions for evolution equations, connected with kinetic models in statistical physics. There is a wide variety of problems where such models are used to obtain reasonable physical as well as numerical results (Fluid Mechanics, Gas Dynamics, Plasma Physics, Nuclear Physics, Turbulence Theory etc.). The classical examples provide the nonlinear Boltzmann equation. Investigation of the long-time behavior of the solutions for the Boltzmann equation gives an approach to the nonlinear fluid dynamic equations. From the viewpoint of dynamical systems, the fluid dynamic equations arise in the theory as a tool to describe an attractor of the kinetic equation.

Inverse Problems for Kinetic and Other Evolution Equations
Inverse Problems for Kinetic and Other Evolution Equations

Author: I︠U︡riĭ Evgenʹevich Anikonov

Publisher: VSP

Published: 2001

Total Pages: 288

ISBN-13: 9789067643450

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This monograph in the "Inverse and Ill-Posed Problems Series deals with methods of studying multidimensional inverse problems for kinetic and other evolution equations, in particular transfer equations. The methods used are applied to concrete inverse problems, especially multidimensional inverse problems applicable in linear and nonlinear statements.A significant part of the book contains formulas and relations for solving inverse problems, including formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, and second order equations.This monograph will be of value and interest to mathematicians, engineers and other specialists dealing with inverse and ill posed problems.

Inverse Problems for Kinetic and Other Evolution Equations
Inverse Problems for Kinetic and Other Evolution Equations

Author: Yu. E. Anikonov

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2014-07-24

Total Pages: 280

ISBN-13: 3110940906

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This monograph deals with methods of studying multidimensional inverse problems for kinetic and other evolution equations, in particular transfer equations. The methods used are applied to concrete inverse problems, especially multidimensional inverse problems applicable in linear and nonlinear statements. A significant part of the book contains formulas and relations for solving inverse problems, including formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, and second order equations.

Nonlinear Evolution Equations
Nonlinear Evolution Equations

Author: Songmu Zheng

Publisher: CRC Press

Published: 2004-07-08

Total Pages: 304

ISBN-13: 0203492226

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Nonlinear evolution equations arise in many fields of sciences including physics, mechanics, and material science. This book introduces some important methods for dealing with these equations and explains clearly and concisely a wide range of relevant theories and techniques. These include the semigroup method, the compactness and monotone operator

Harmonic Analysis Method for Nonlinear Evolution Equations, I
Harmonic Analysis Method for Nonlinear Evolution Equations, I

Author: Baoxiang Wang

Publisher: World Scientific

Published: 2011

Total Pages: 298

ISBN-13: 9814360740

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1. Fourier multiplier, function space [symbol]. 1.1. Schwartz space, tempered distribution, Fourier transform. 1.2. Fourier multiplier on L[symbol]. 1.3. Dyadic decomposition, Besov and Triebel spaces. 1.4. Embeddings on X[symbol]. 1.5. Differential-difference norm on [symbol]. 1.6. Homogeneous space [symbol] 1.7. Bessel (Riesz) potential spaces [symbol]. 1.8. Fractional Gagliardo-Nirenberg inequalities -- 2. Navier-Stokes equation. 2.1. Introduction. 2.2. Time-space estimates for the heat semi-group. 2.3. Global well-posedness in L[symbol] of NS in 2D. 2.4. Well-posedness in L[symbol] of NS in higher dimensions. 2.5. Regularity of solutions for NS -- 3. Strichartz estimates for linear dispersive equations. 3.1. [symbol] estimates for the dispersive semi-group. 3.2. Strichartz inequalities : dual estimate techniques. 3.3. Strichartz estimates at endpoints -- 4. Local and global wellposedness for nonlinear dispersive equations. 4.1. Why is the Strichartz estimate useful. 4.2. Nonlinear mapping estimates in Besov spaces. 4.3. Critical and subcritical NLS in H[symbol]. 4.4. Global wellposedness of NLS in L[symbol] and H[symbol]. 4.5. Critical and subcritical NLKG in H[symbol]. 5. The low regularity theory for the nonlinear dispersive equations. 5.1. Bourgain space. 5.2. Local smoothing effect and maximal function estimates. 5.3. Bilinear estimates for KdV and local well-posedness. 5.4. Local well-posedness for KdV in H[symbol]. 5.5. I-method. 5.6. Schrodinger equation with derivative. 5.7. Some other dispersive equations -- 6. Frequency-uniform decomposition techniques. 6.1. Why does the frequency-uniform decomposition work. 6.2. Frequency-uniform decomposition, modulation spaces. 6.3. Inclusions between Besov and modulation spaces. 6.4. NLS and NLKG in modulation spaces. 6.5. Derivative nonlinear Schrodinger equations -- 7. Conservations, Morawetz' estimates of nonlinear Schrodinger equations. 7.1. Nother's theorem. 7.2. Invariance and conservation law. 7.3. Virial identity and Morawetz inequality. 7.4. Morawetz' interaction inequality. 7.5. Scattering results for NLS -- 8. Boltzmann equation without angular cutoff. 8.1. Models for collisions in kinetic theory. 8.2. Basic surgery tools for the Boltzmann operator. 8.3. Properties of Boltzmann collision operator without cutoff. 8.4 Regularity of solutions for spatially homogeneous case

Nonlinear Evolution Equations and Dynamical Systems
Nonlinear Evolution Equations and Dynamical Systems

Author: Sandra Carillo

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 247

ISBN-13: 3642840396

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Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. Except for a few review papers, the 40 contributions are intentially brief to give only the gist of the methods, proofs, etc. including references to the relevant litera- ture. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior resercher as well as the colleague just entering the field. Keypoints treated are: i) integrable systems in multidimensions and associated phenomenology ("dromions"); ii) criteria and tests of integrability (e.g., Painlev test); iii) new developments related to the scattering transform; iv) algebraic approaches to integrable systems and Hamiltonian theory (e.g., connections with Young-Baxter equations and Kac-Moody algebras); v) new developments in mappings and cellular automata, vi) applications to general relativity, condensed matter physics, and oceanography.

Lectures on Nonlinear Evolution Equations
Lectures on Nonlinear Evolution Equations

Author: Reinhard Racke

Publisher: Birkhäuser

Published: 2015-08-31

Total Pages: 315

ISBN-13: 3319218735

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This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.

The Vector Field Method and Its Applications to Nonlinear Evolution Equations
The Vector Field Method and Its Applications to Nonlinear Evolution Equations

Author: Leonardo Enrique Abbrescia

Publisher:

Published: 2020

Total Pages: 229

ISBN-13:

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The vector field method was introduced in the 1980s by Sergiu Klainerman to analyze the decay properties of the linear wave equation. Since its historical debut, the vector field method has been at the forefront of several breakthrough results including the global stability of Minkowski space, the dynamical formation of black holes, and shock formation in 3D compressible fluids.This work showcases how the vector field method can be used in a systematic way to derive a priori estimates for nonlinear evolution equations. For nonlinear dispersive equations, these estimates can be married to the decay properties enjoyed by the solutions to derive quantitative asymptotics. This is done in this work through the lens of three concrete problems: a nonlocal kinetic model, the wave maps equation, and the relativistic membrane equation. For the kinetic model, the vector field method is paired with dispersive decay properties of the spatial density to prove global wellposedness of small data. This can be interpreted physically as "stability" of the trivial solution. For the wave maps equation, a stability result is proven for a "non-trivial" ODE geodesic solution. For the relativistic membrane equation, the vector field method is used to prove stability of large simple-traveling-waves. For the wave map and membrane equations, we intimately use several structural properties known as null conditions that preclude singular behavior.

Nonlinear Evolution Equations
Nonlinear Evolution Equations

Author: Boling Guo

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2019-11-05

Total Pages: 369

ISBN-13: 3110614782

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Nonlinear Evolution Equation presents state-of-the-art theories and results on nonlinear evolution equation, showing related mathematical methods and applications. The basic concepts and research methods of infinite dimensional dynamical systems are discussed in detail. The unique combination of mathematical rigor and physical background makes this work an essential reference for researchers and students in applied mathematics and physics.