Dual Variational Approach to Nonlinear Diffusion Equations
Dual Variational Approach to Nonlinear Diffusion Equations

Author: Gabriela Marinoschi

Publisher: Springer Nature

Published: 2023-03-28

Total Pages: 223

ISBN-13: 3031245830

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This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical models to various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well.

Semigroup Approach To Nonlinear Diffusion Equations
Semigroup Approach To Nonlinear Diffusion Equations

Author: Viorel Barbu

Publisher: World Scientific

Published: 2021-09-23

Total Pages: 221

ISBN-13: 981124653X

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This book is concerned with functional methods (nonlinear semigroups of contractions, nonlinear m-accretive operators and variational techniques) in the theory of nonlinear partial differential equations of elliptic and parabolic type. In particular, applications to the existence theory of nonlinear parabolic equations, nonlinear Fokker-Planck equations, phase transition and free boundary problems are presented in details. Emphasis is put on functional methods in partial differential equations (PDE) and less on specific results.

Nonlinear Diffusion Equations and Their Equilibrium States II
Nonlinear Diffusion Equations and Their Equilibrium States II

Author: W.-M. Ni

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 364

ISBN-13: 1461396085

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In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = ~U + f(u). Here ~ denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium ~u+f(u)=O. Particular cases arise, for example, in population genetics, the physics of nu clear stability, phase transitions between liquids and gases, flows in porous media, the Lend-Emden equation of astrophysics, various simplified com bustion models, and in determining metrics which realize given scalar or Gaussian curvatures. In the latter direction, for example, the problem of finding conformal metrics with prescribed curvature leads to a ground state problem involving critical exponents. Thus not only analysts, but geome ters as well, can find common ground in the present work. The corresponding mathematical problem is to determine how the struc ture of the nonlinear function f(u) influences the behavior of the solution.

Solutions Of Nonlinear Differential Equations: Existence Results Via The Variational Approach
Solutions Of Nonlinear Differential Equations: Existence Results Via The Variational Approach

Author: Lin Li

Publisher: World Scientific

Published: 2016-04-15

Total Pages: 362

ISBN-13: 9813108622

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Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics (including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis).In our first chapter, we gather the basic notions and fundamental theorems that will be applied throughout the chapters. While many of these items are easily available in the literature, we gather them here both for the convenience of the reader and for the purpose of making this volume somewhat self-contained. Subsequent chapters deal with how variational methods can be used in fourth-order problems, Kirchhoff problems, nonlinear field problems, gradient systems, and variable exponent problems. A very extensive bibliography is also included.

Nonlinear Diffusion Equations and Their Equilibrium States I
Nonlinear Diffusion Equations and Their Equilibrium States I

Author: W.-M. Ni

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 359

ISBN-13: 1461396050

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In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = D.u + f(u). Here D. denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium D.u + f(u) = o. Particular cases arise, for example, in population genetics, the physics of nu clear stability, phase transitions between liquids and gases, flows in porous media, the Lend-Emden equation of astrophysics, various simplified com bustion models, and in determining metrics which realize given scalar or Gaussian curvatures. In the latter direction, for example, the problem of finding conformal metrics with prescribed curvature leads to a ground state problem involving critical exponents. Thus not only analysts, but geome ters as well, can find common ground in the present work. The corresponding mathematical problem is to determine how the struc ture of the nonlinear function f(u) influences the behavior of the solution.

Nonlinear Diffusion Equations
Nonlinear Diffusion Equations

Author: Zhuoqun Wu

Publisher: World Scientific

Published: 2001

Total Pages: 521

ISBN-13: 9810247184

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Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations.This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.

Self-dual Partial Differential Systems and Their Variational Principles
Self-dual Partial Differential Systems and Their Variational Principles

Author: Nassif Ghoussoub

Publisher: Springer Science & Business Media

Published: 2008-11-11

Total Pages: 352

ISBN-13: 0387848967

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This text is intended for a beginning graduate course on convexity methods for PDEs. The generality chosen by the author puts this under the classification of "functional analysis". The book contains new results and plenty of examples and exercises.

Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations
Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations

Author: Donald J. Estep

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 125

ISBN-13: 0821820729

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This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally. We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.

Handbook of Differential Equations: Evolutionary Equations
Handbook of Differential Equations: Evolutionary Equations

Author: C.M. Dafermos

Publisher: Elsevier

Published: 2008-10-06

Total Pages: 609

ISBN-13: 0080931979

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The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts.- Review of new results in the area- Continuation of previous volumes in the handbook series covering Evolutionary PDEs- Written by leading experts