A.M. Meirmanov

Evolution Equations A.M. Meirmanov

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.