Tujin Kim - Equations of Motion for Incompressible Viscous Fluids: With Mixed Boundary Conditions

The Advances in Mathematical Fluid Mechanics series is a forum for the publication of high-quality, peer-reviewed research monographs and edited collections on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations and other significant viscous and inviscid fluid models. Titles in this series consider theoretical, numerical, and computational methods, as well as applications to science and engineering. Works in related areas of mathematics that have a direct bearing on fluid mechanics are also welcome. All manuscripts are peer-reviewed to meet the highest standards of scientific literature.

More information about this series at http://www.springer.com/series/5032

This book is published under the imprint Birkhuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AGThis book is published under the imprint Birkhuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

The study of the equations appearing in fluid mechanics is a core topic in the fields of partial differential equations, fluid mechanics and applied sciences. Lots of real phenomena of fluid are described by those equations with different kinds of boundary conditions. We are concerned with various boundary conditions, and nowadays for mathematical analysis of phenomena of fluid, the fluid equations with new boundary conditions are studied. In practice, we are concerned with mixture of various boundary conditions. Now there are many papers dealing with fluid equations with mixed boundary conditions; however, almost all monographs deal with the equations with Dirichlet boundary condition and a few monographs deal with mixture of Dirichlet and stress or Dirichlet and a kind of friction boundary conditions.

The purpose of this book is to introduce the recent results and research methods in the field of fluid equations with mixed boundary conditions via one book. For readers broadly ranging from students to engineers and mathematicians involved in the fluid equations, first the fluid equations and the boundary conditions for the Navier-Stokes equations are outlined in the level of senior university students. Then, the recent results for the Navier-Stokes equations and the equations of motion for fluid conducting heat with the mixed boundary conditions originated mainly from authors results and the first authors lectures for postgraduate students in the Institute of Mathematics, State Academy of Sciences, DPR Korea are described in the level of postgraduate students.

The book consists of 10 chapters.

Trying to make our book self-contained, in Chap..

In Chap. , we first show how the Navier-Stokes equations and the equations of motion for fluid under consideration of heat are derived from the fundamental physical laws. Though the aim of this book is to study the equations of motion for the incompressible fluids, the equations for compressible fluids are discussed together to widen readers understanding. Next, we outline some boundary conditions for the Navier-Stokes equations, which would be helpful for readers to understand the meaning of the boundary conditions in this book and to improve the ability to apply these in practice. Then, we consider the widely used existing three kinds of variational formulations for the Navier-Stokes problems with mixed boundary conditions. Also, equivalence between the variational formulations and the original PDE problems is discussed, which is a preparation to understand equivalence between PDE problems with more complicated boundary conditions and their variational formulations in the subsequent chapters. This chapter can serve as an introduction for the students and postgraduate students in the fields of equations describing fluids, mathematical modeling and numerical simulation.