Mathematical tools for the study of the incompressible Navier-Stokes equations and related models
In this interesting and very well-written book, the authors are concerned with the mathematical analysis of Navier-Stokes equations and related problems and models. The book is divided into seven chapters followed by two appendices, a list of references and an index. In the first chapter, the authors present the equations of fluid mechanics and related models that are studied in the forthcoming chapters. The second chapter reviews the main results in functional analysis that are used in the book, such as the weak and weak-\(*\) convergence in Banach spaces, the distribution theory, compactness results and some results from spectral theory. In the third chapter, the authors provide an introduction to Sobolev spaces in Lipschitz domains in \({\mathbb R}^d\). The fourth chapter examines the steady Stokes equations. The authors present a complete proof of Nečas inequality, which is used in the second section of this chapter for the study of some relations between gradient fields and divergence-free vector fields. Certain properties of divergence and curl operators are also presented. In the fifth section of Chapter four, the authors investigate the Stokes problem with Dirichlet boundary conditions, and show the well-posedness of the Stokes system. They also study the unbounded Stokes operator and present elliptic regularity for the Stokes problem. In the last three sections of Chapter four, the authors deal with Stokes equations with other boundary conditions, such as Neumann boundary conditions, interface conditions, and a boundary condition in terms of vorticity field. They prove existence, uniqueness and regularity results for these problems. The fifth chapter is devoted to steady and unsteady equations for homogeneous fluids. The authors prove the existence, uniqueness and regularity of solutions to the incompressible Navier-Stokes equations in bounded domains in \({\mathbb R}^d\), for \(d=2,3\). The first section provides an introduction to weak solutions, while the second section deals with the existence and uniqueness of strong solutions. The last section of this chapter provides the existence and uniqueness of solutions to the steady Navier-Stokes equations with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability results are also presented. In the sixth chapter, the authors study the unsteady incompressible Navier-Stokes equations for nonhomogeneous fluids. At the beginning, they investigate the weak solutions of the transport equations. Then the authors provide a deep presentation of the Di Parma-Lions theory of renormalized solutions for such equations. Moreover, the authors show the global existence of weak solutions for initial and boundary value problems for the Navier-Stokes system. The seventh chapter is concerned with outflow boundary conditions, the related existence and uniqueness results, and Dirichlet boundary conditions. The proofs are obtained by a penalty method. The book ends with two appendices. The first one deals with differential operators, while the second one gives an overview of thermodynamics. This book is a valuable contribution to the theory of Navier-Stokes equations and related models. It contains important existence, uniqueness and stability results, together with a rigorous analysis of all treated problems. It is warmly recommended to all researchers interested in modern nonlinear problems in fluid mechanics.
scientific article; zbMATH DE number 6106780