The objective of this book is to report the results of investigations made by the authors into certain hydrodynamical models with nonlinear systems of partial differential equations.The investigations involve the results concerning Navier-Stokes equations of viscous heat-conductive gas, incompressible nonhomogeneous fluid and filtration of multi-phase mixture in a porous medium. The correctness of the initial boundary-value problems and the qualitative properties of solutions are also considered. The book is written for those who are interested in the theory of nonlinear partial differential equations and their applications in mechanics.

For the past several decades, the study of free boundary problems has been a very active subject of research occurring in a variety of applied sciences. What these problems have in common is their formulation in terms of suitably posed initial and boundary value problems for nonlinear partial differential equations. Such problems arise, for example, in the mathematical treatment of the processes of heat conduction, filtration through porous media, flows of non-Newtonian fluids, boundary layers, chemical reactions, semiconductors, and so on. The growing interest in these problems is reflected by the series of meetings held under the title "Free Boundary Problems: Theory and Applications" (Ox ford 1974, Pavia 1979, Durham 1978, Montecatini 1981, Maubuisson 1984, Irsee 1987, Montreal 1990, Toledo 1993, Zakopane 1995, Crete 1997, Chiba 1999). From the proceedings of these meetings, we can learn about the different kinds of mathematical areas that fall within the scope of free boundary problems. It is worth mentioning that the European Science Foundation supported a vast research project on free boundary problems from 1993 until 1999. The recent creation of the specialized journal Interfaces and Free Boundaries: Modeling, Analysis and Computation gives us an idea of the vitality of the subject and its present state of development. This book is a result of collaboration among the authors over the last 15 years.

Nonlinear Evolution Equation covers the proceedings of the Symposium by the same title, conducted by the Mathematics Research Center at the University of Wisconsin, Madison on October 17-19, 1977. This book is divided into 13 chapters and begins with reviews of the uniqueness of solution to systems of conservation laws and the computational aspects of Glimm's method. The next chapters examine the theoretical and practical aspects of Boltzmann, Navier-Stokes, and evolution equations. These topics are followed by discussions of the practical applications of Trotter's product formula for some nonlinear semigroups and the finite time blow-up in nonlinear problems. The closing chapters deal with a Hamiltonian approach to the K-dV and other equations, along with a variational method for finding periodic solutions of differential equations. This book will prove useful to mathematicians and engineers.

On November 3, 2005, Alexander Vasil’evich Kazhikhov left this world, untimely and unexpectedly. He was one of the most in?uential mathematicians in the mechanics of ?uids, and will be remembered for his outstanding results that had, and still have, a c- siderablysigni?cantin?uenceinthe?eld.Amonghis manyachievements,werecall that he was the founder of the modern mathematical theory of the Navier-Stokes equations describing one- and two-dimensional motions of a viscous, compressible and heat-conducting gas. A brief account of Professor Kazhikhov’s contributions to science is provided in the following article “Scienti?c portrait of Alexander Vasil’evich Kazhikhov”. This volume is meant to be an expression of high regard to his memory, from most of his friends and his colleagues. In particular, it collects a selection of papers that represent the latest progress in a number of new important directions of Mathematical Physics, mainly of Mathematical Fluid Mechanics. These papers are written by world renowned specialists. Most of them were friends, students or colleagues of Professor Kazhikhov, who either worked with him directly, or met him many times in o?cial scienti?c meetings, where they had the opportunity of discussing problems of common interest.

This monograph is concerned with free-boundary problems of partial differential equations arising in the physical sciences and in engineering. The existence and uniqueness of solutions to the Hele-Shaw problem are derived and techniques to deal with the Muskat problem are discussed. Based on these, mathematical models for the dynamics of cracks in underground rocks and in-situ leaching are developed. Contents Introduction The Hele–Shaw problem A joint motion of two immiscible viscous fluids Mathematical models of in-situ leaching Dynamics of cracks in rocks Elements of continuum mechanics

This edited volume gathers selected, peer-reviewed contributions presented at the fourth International Conference on Differential & Difference Equations Applications (ICDDEA), which was held in Lisbon, Portugal, in July 2019. First organized in 2011, the ICDDEA conferences bring together mathematicians from various countries in order to promote cooperation in the field, with a particular focus on applications. The book includes studies on boundary value problems; Markov models; time scales; non-linear difference equations; multi-scale modeling; and myriad applications.