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Evolution Equations and Lagrangian Coordinates : Groningen-Amsterdam Studies in Semantics (Grass) - Anvarbek M. Meirmanov

Evolution Equations and Lagrangian Coordinates

Groningen-Amsterdam Studies in Semantics (Grass)

Hardcover Published: 3rd March 1997
ISBN: 9783110148756
Number Of Pages: 324
For Ages: 22+ years old

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This book presents recent results of the investigation of evolution equations by means of the method of Lagrangian coordinates. This monograph is concerned with the investigation of evolution equations and boundary value problems for such equations via the method of Lagrangian coordinates. There are two basic methods for describing the motions of continua: those of Euler and Lagrange. They both reflect our physical understanding about the process and are widely used in continuum mechanics. However, the Euler and Lagrange models corresponding to the same physical process are of different mathematical nature. This provides a possibility to consider an abstract evolution equation. Given an evolution equation - whatever its physical applications may be - it can be interpreted as conservation law for the motion of some fictitious continuum written in Euler variables and can then be given the alternative formulation via Lagrangian coordinates. In this book this ides developed in several directions. It is used to analyze the solvability of Verigin's problem arising from the theory of filtration through a porous soil and to study the one-phase Stefan problem. New special symmetry properties of evolution equations ar derived and new partial exact solutions to various equations and problems are found. The method of Lagrangian coordinates is particularly useful for dealing with problems which involve free boundaries or interfaces. In this framework, the authors study the qualitative properties (regularity and asymptotic behaviour) of the interfaces displayed by the solutions to degenerate parabolic equations of a wide class. Special emphasis is paid to the so-called porous medium equation (both in one- and multi-dimensional setting) and its generalizations. This volume is of interest to analysts in particular specialists in partial differential equations as well as physicists and engineers.

The Verigin problem
Review of resultp. 1
Filtration in porous soilp. 2
Formulation of the problemp. 4
Self-similar solutions; Stefan's problem as limit case of Verigin's problemp. 6
One-dimensional problem: main statements and formulation of resultsp. 9
Proofs of Theorems 5.1 and 5.2: Verigin's problems with given mass flux on the known boundaries and the Cauchy-Verigin problemp. 12
Proof of Theorem 5.3: Verigin's problem with the given pressure on the known boundariesp. 33
Equivalence transformations of evolution equations
Main ideas. A historical surveyp. 42
Reciprocity transformations of second-order equationsp. 47
Hidden symmetry of evolution equationsp. 51
Linearization by means of Lagrangian coordinatesp. 62
Lagrange-invariants equationsp. 65
Equations with spherical and cylindrical symmetriesp. 69
Equivalence transformations for higher-order equations and systems of equationsp. 79
A remarkable equation of nonlinear heat conductionp. 89
The one-phase Stefan problem: explicit solutions with functional arbitrarinessp. 94
One-dimensional parabolic equations
Introductionp. 104
Lagrangian coordinates in one-dimensional evolution equationsp. 121
Analysis of the problem in Lagrangian terminologyp. 130
Uniform estimates: The inverse transformationp. 137
Some starting properties of the interfacep. 143
Estimates for the time derivative and the higher-order derivativesp. 146
The interface regularityp. 158
Regularity of interfaces for a generalized porous medium equationp. 161
Axially symmetrical solutions of the porous medium equation: long-time asymptotic behaviorp. 185
A non-classical problem for a degenerate parabolic equation: uniqueness of unbounded solutionsp. 204
The Stefan problem with degeneracy at the free boundary: example of exact solutionp. 207
Parabolic equations in several space dimensions
Review of resultsp. 210
Langrangian coordinates in the one-phase Stefan problemp. 212
Correctness of the linear modelp. 217
Similarity solutions of the Stefan problemp. 222
Solvability of the nonlinear problemp. 237
Canonical Lagrangian coordinatesp. 244
Boussinesq's equation in filtration theoryp. 250
Local regularity of interfacesp. 270
Bibliographyp. 295
Notationp. 309
Indexp. 310
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783110148756
ISBN-10: 3110148757
Series: Groningen-Amsterdam Studies in Semantics (Grass)
Audience: Professional
For Ages: 22+ years old
Format: Hardcover
Language: English
Number Of Pages: 324
Published: 3rd March 1997
Country of Publication: DE
Dimensions (cm): 24.44 x 17.42  x 2.24
Weight (kg): 0.68