Evolution equations and Lagrangian coordinates

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Evolution equations and Lagrangian coordinates

by A.M. Meirmanov, V.V. Pukhnachov, S.I. Shmarev

（De Gruyter expositions in mathematics, 24）

Walter de Gruyter & Company, 1997

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Includes bibliographical references and index

Description and Table of Contents

Description

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Table of Contents

  • The Verigin problem: review of results
  • filtration in a porous soil
  • formulation of the problem
  • self-similar solutions
  • Stefan's problem as a limit case of Verigin's problem
  • one-dimensional problem - main statements and formulation of results
  • proofs of theorems 5.1, 5.2 - Verigin's problem with the given mass flux on the known boundaries and the Cauchy-Verigin problem
  • proof of theorem 5.3
  • Verigin's problem with the given pressure on the known boundaries. equivalence transformations of evolution equations: main ideas
  • a historical survey
  • reciprocity transformations of second-order equations
  • hidden symmetry of evolution equations
  • linearization by means of Lagrangian coordinates
  • Lagrange-invariant equations
  • equations with spherical and cylindrical symmetries
  • equivalence transformations for higher-order equations and systems of equations
  • a remarkable equation of nonlinear heat conduction
  • the on-phase Stefan problem - explicit solutions with functional arbitrariness. One dimensional parabolic equations: introduction
  • Lagrangian coordinates in one-dimensional evolution equations
  • analysis of the problem in Langragian terminology
  • uniform estimates
  • the inverse transformation
  • some starting properties of the interface
  • estimates for the time derivatives and the higher-order derivatives
  • the interface regularity
  • regularity of interfaces for a generalized porous medium equation
  • axially symmetrical solutions of the porous medium equation - long-time asymptotic behaviour
  • a non-classical problem for a degenerate parabolic equation - uniqueness of unbounded solutions
  • the Stefan problem with degeneracy at the free boundary - example of exact solution. Parabolic equations in several space dimensions: review of results
  • Langragian coordinates in the one-phase Stefan problem
  • correctness of the linear model
  • similarity solutions of the Stefan problem
  • solvability of the nonlinear problem
  • canonical Lagrangian coordinates
  • Boussinesq's equation in filtration theory
  • local regularity of interfaces
  • bibliography
  • notation.

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