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On The Local Behaviour Of Solutions Of Degenerate Parabolic Equations With Measurable Coefficients
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Book Synopsis On the Local Behaviour of Solutions of Degenerate Parabolic Equations with Measurable Coefficients by : E. DiBenedetto
Download or read book On the Local Behaviour of Solutions of Degenerate Parabolic Equations with Measurable Coefficients written by E. DiBenedetto and published by . This book was released on 1984 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt: Locally bounded weak solutions of degenerate parabolic equations are proven to be locally hoelder continuous. Hoelder estimates are also derived up to the boundary for both Dirichlet data and (non-linear) variational data. Via a counterexample it is shown that non-negative solutions, in general, do not satisfy the parabolic version of the Harnack inequality.
Author :Emmanuele DiBenedetto Publisher :Springer Science & Business Media ISBN 13 :1461208955 Total Pages :402 pages Book Rating :4.4/5 (612 download)
Book Synopsis Degenerate Parabolic Equations by : Emmanuele DiBenedetto
Download or read book Degenerate Parabolic Equations written by Emmanuele DiBenedetto and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Evolved from the author's lectures at the University of Bonn's Institut für angewandte Mathematik, this book reviews recent progress toward understanding of the local structure of solutions of degenerate and singular parabolic partial differential equations.
Book Synopsis Local behaviour of solutions to parabolic equations by :
Download or read book Local behaviour of solutions to parabolic equations written by and published by . This book was released on 1987 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis On the Local Behaviour of Solutions of Singular Parabolic Equations by : Vincenzo Vespri
Download or read book On the Local Behaviour of Solutions of Singular Parabolic Equations written by Vincenzo Vespri and published by . This book was released on 1990 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonlinear Diffusion Equations by : Zhuoqun Wu
Download or read book Nonlinear Diffusion Equations written by Zhuoqun Wu and published by World Scientific. This book was released on 2001 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Contents: Newtonian Filtration Equations: Existence and Uniqueness of Solutions: One Dimensional Case; Existence and Uniqueness of Solutions: Higher Dimensional Case; Regularity of Solutions: One Dimensional Case; Regularity of Solutions: Higher Dimensional Case; Properties of the Free Boundary: One Dimensional Case; Properties of the Free Boundary: Higher Dimensional Case; Initial Trace of Solutions; Other Problems; Non-Newtonian Filtration Equations: Existence of Solutions; Harnack Inequality and Initial Trace of Solutions; Regularity of Solutions; Uniqueness of Solutions; Properties of the Free Boundary; Other Problems; General Quasilinear Equations of Second Order: Weakly Degenerate Equations in One Dimension; Weakly Degenerate Equations in Higher Dimension; Strongly Degenerate Equations in One Dimension; Degenerate Equations in Higher Dimension without Terms of Lower Order; General Strongly Degenerate Equations in Higher Dimension; Classes BV and BV x; Nonlinear Diffusion Equations of Higher Order: Similarity Solutions of a Fourth Order Equation; Equations with Double-Degeneracy; CahnOCoHilliard Equation with Constant Mobility; CahnOCoHilliard Equations with Positive Concentration Dependent Mobility; Thin Film Equation; CahnOCoHilliard Equation with Degenerate Mobility. Readership: Researchers, lecturers and graduate students in the fields of analysis and differential equations, mathematical physics and fluid mechanics."
Book Synopsis Elliptic Regularity Theory by Approximation Methods by : Edgard A. Pimentel
Download or read book Elliptic Regularity Theory by Approximation Methods written by Edgard A. Pimentel and published by Cambridge University Press. This book was released on 2022-06-30 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs – such as the Krylov–Safonov and Evans–Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ – and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.
Book Synopsis The Regularity of General Parabolic Systems with Degenerate Diffusion by : Verena Bögelein
Download or read book The Regularity of General Parabolic Systems with Degenerate Diffusion written by Verena Bögelein and published by American Mathematical Soc.. This book was released on 2013-01-28 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\partial_t u - \mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.
Book Synopsis Harnack Inequalities and Nonlinear Operators by : Vincenzo Vespri
Download or read book Harnack Inequalities and Nonlinear Operators written by Vincenzo Vespri and published by Springer Nature. This book was released on 2021-05-29 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book contains two contributions about the work of Emmanuele DiBenedetto and a selection of original papers. The authors are some of the main experts in Harnack’s inequalities and nonlinear operators. These papers are part of the contributions presented during the conference to celebrate the 70th birthday of Prof. Emmanuele DiBenedetto, which was held at “Il Palazzone” in Cortona from June 18th to 24th, 2017. The papers are focused on current research topics regarding the qualitative properties of solutions, connections with calculus of variations, Harnack inequality and regularity theory. Some papers are also related to various applications. Many of the authors have shared with Prof. DiBenedetto an intense scientific and personal collaboration, while many others have taken inspiration from and further developed his field of research. The topics of the conference are certainly of great interest for the international mathematical community.
Author :Emmanuele DiBenedetto Publisher :Springer Science & Business Media ISBN 13 :1461415845 Total Pages :287 pages Book Rating :4.4/5 (614 download)
Book Synopsis Harnack's Inequality for Degenerate and Singular Parabolic Equations by : Emmanuele DiBenedetto
Download or read book Harnack's Inequality for Degenerate and Singular Parabolic Equations written by Emmanuele DiBenedetto and published by Springer Science & Business Media. This book was released on 2011-11-13 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements, the issue of the Harnack inequality for non-negative solutions to these equations, both of p-Laplacian and porous medium type, while raised by several authors, has remained basically open. Recently considerable progress has been made on this issue, to the point that, except for the singular sub-critical range, both for the p-laplacian and the porous medium equations, the theory is reasonably complete. It seemed therefore timely to trace a comprehensive overview, that would highlight the main issues and also the problems that still remain open. The authors give a comprehensive treatment of the Harnack inequality for non-negative solutions to p-laplace and porous medium type equations, both in the degenerate (p/i”2 or im/i”1) and in the singular range (1“ip/i2 or 0“im/i
Book Synopsis Parabolic Systems with Polynomial Growth and Regularity by : Frank Duzaar
Download or read book Parabolic Systems with Polynomial Growth and Regularity written by Frank Duzaar and published by American Mathematical Soc.. This book was released on 2011 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems $ u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,$ under the main assumption of polynomial growth at rate $p$ i.e. $ a(x,t,u,Du) \leq L(1+ Du ^{p-1}), p \geq 2.$ They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.
Book Synopsis Elliptic and Parabolic Problems by : C Bandle
Download or read book Elliptic and Parabolic Problems written by C Bandle and published by CRC Press. This book was released on 2020-11-26 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Research Note presents some recent advances in various important domains of partial differential equations and applied mathematics including equations and systems of elliptic and parabolic type and various applications in physics, mechanics and engineering. These topics are now part of various areas of science and have experienced tremendous development during the last decades. -------------------------------------
Book Synopsis Continuity of Solutions of Degenerate Parabolic Equations by : Paul Sacks
Download or read book Continuity of Solutions of Degenerate Parabolic Equations written by Paul Sacks and published by . This book was released on 1981 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Second Order Parabolic Differential Equations by : Gary M Lieberman
Download or read book Second Order Parabolic Differential Equations written by Gary M Lieberman and published by World Scientific. This book was released on 1996-11-06 with total page 462 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to the general theory of second order parabolic differential equations, which model many important, time-dependent physical systems. It studies the existence, uniqueness, and regularity of solutions to a variety of problems with Dirichlet boundary conditions and general linear and nonlinear boundary conditions by means of a priori estimates. The first seven chapters give a description of the linear theory and are suitable for a graduate course on partial differential equations. The last eight chapters cover the nonlinear theory for smooth solutions. They include much of the author's research and are aimed at researchers in the field. A unique feature is the emphasis on time-varying domains.
Book Synopsis The Porous Medium Equation by : Juan Luis Vazquez
Download or read book The Porous Medium Equation written by Juan Luis Vazquez and published by Oxford University Press. This book was released on 2007 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed at research students and academics in mathematics and engineering, as well as engineering specialists, this book provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation.
Book Synopsis Attractors for Degenerate Parabolic Type Equations by : Messoud Efendiev
Download or read book Attractors for Degenerate Parabolic Type Equations written by Messoud Efendiev and published by American Mathematical Soc.. This book was released on 2013-09-26 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the long-time behavior of solutions of degenerate parabolic dissipative equations arising in the study of biological, ecological, and physical problems. Examples include porous media equations, -Laplacian and doubly nonlinear equations, as well as degenerate diffusion equations with chemotaxis and ODE-PDE coupling systems. For the first time, the long-time dynamics of various classes of degenerate parabolic equations, both semilinear and quasilinear, are systematically studied in terms of their global and exponential attractors. The long-time behavior of many dissipative systems generated by evolution equations of mathematical physics can be described in terms of global attractors. In the case of dissipative PDEs in bounded domains, this attractor usually has finite Hausdorff and fractal dimension. Hence, if the global attractor exists, its defining property guarantees that the dynamical system reduced to the attractor contains all of the nontrivial dynamics of the original system. Moreover, the reduced phase space is really "thinner" than the initial phase space. However, in contrast to nondegenerate parabolic type equations, for a quite large class of degenerate parabolic type equations, their global attractors can have infinite fractal dimension. The main goal of the present book is to give a detailed and systematic study of the well-posedness and the dynamics of the semigroup associated to important degenerate parabolic equations in terms of their global and exponential attractors. Fundamental topics include existence of attractors, convergence of the dynamics and the rate of convergence, as well as the determination of the fractal dimension and the Kolmogorov entropy of corresponding attractors. The analysis and results in this book show that there are new effects related to the attractor of such degenerate equations that cannot be observed in the case of nondegenerate equations in bounded domains. This book is published in cooperation with Real Sociedad Matemática Española (RSME).
Book Synopsis The Method of Intrinsic Scaling by : José Miguel Urbano
Download or read book The Method of Intrinsic Scaling written by José Miguel Urbano and published by Springer Science & Business Media. This book was released on 2008-05-20 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: This set of lectures, which had its origin in a mini course delivered at the Summer Program of IMPA (Rio de Janeiro), is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular PDEs.In the first part, the theory is presented from scratch for the model case of the degenerate p-Laplace equation. The second part deals with three applications of the theory to relevant models arising from flows in porous media and phase transitions.
Book Synopsis Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients by : Paweł Strzelecki
Download or read book Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients written by Paweł Strzelecki and published by . This book was released on 1990 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt:
