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**Mathematical models in boundary layer theory.**
*(English)*
Zbl 0928.76002

Applied Mathematics and Mathematical Computation. 15. Boca Raton, FL: Chapman & Hall/ CRC. 516 p. (1999).

The monograph is devoted to the mathematical theory of the boundary layer approach for fluids with small viscosity, flowing about a solid body. These methods are very useful in the nonlinear theory of viscous and electrically conducting flows, in the theory of heat and mass transfer, and in the dynamics of reactive and multi-phase media. The presented approach is devoted to the investigation of such topics as well-posedness of various boundary value problems for the boundary layer system, existence and uniqueness of solutions in certain classes of functions, as well as the problem of stability of solutions with respect to perturbations of given quantities. Another group of problems deals with the qualitative behavior of solutions and their asymptotics. Finally, of great importance are the methods, presented in the monograph, for an approximate solution of the Prandtl system and for subsequent evaluation of the rate of convergence of the approximations to the exact solution.

The book consists of ten chapters. Chapter 1 is of introductory character; here the equations of the boundary layer are derived from the Navier-Stokes system and the principal boundary value problems are stated. Then stationary boundary layers are described using both von Mises and Crocco variables. The nonstationary boundary layer is analyzed in the axially symmetric and time-periodic cases. Also, stability of solutions of the nonstationary Prandtl system is discussed and the existence of solutions along with their regularity properties is proved by the line method in time variable. Chapter 5 is devoted to the formation of boundary layers in the case of gradual acceleration and for a body that suddenly starts to move. In chapter 6 finite difference schemes are constructed for two principal problems considered earlier by other means: continuation problem and Prandtl system for axially symmetric flow. Then two types of problems for the Prandtl system with the transmission conditions on a certain surface are considered: the boundary layer near a wall through which an another fluid is injected, and the problem for two mixed fluids. After that the applications of the boundary layer theory to non-Newtonian flows and to magnetohydrodynamics are considered. By means of the method of homogenization, chapter 10 treats the problems for Prandtl system with rapidly oscillating injection and suction and the MHD problem in a rapidly oscillating magnetic field. The monograph is finished by stating some open problems in the boundary layer theory.

The level of this interesting book is very high, but clear. The exposition is comprehensive, and theorems and lemmas are thoroughly proved. It seems that the authors of this monograph are the first to undertake such a successive and systematic exposition of the main mathematical results and methods in the boundary layer theory.

The book consists of ten chapters. Chapter 1 is of introductory character; here the equations of the boundary layer are derived from the Navier-Stokes system and the principal boundary value problems are stated. Then stationary boundary layers are described using both von Mises and Crocco variables. The nonstationary boundary layer is analyzed in the axially symmetric and time-periodic cases. Also, stability of solutions of the nonstationary Prandtl system is discussed and the existence of solutions along with their regularity properties is proved by the line method in time variable. Chapter 5 is devoted to the formation of boundary layers in the case of gradual acceleration and for a body that suddenly starts to move. In chapter 6 finite difference schemes are constructed for two principal problems considered earlier by other means: continuation problem and Prandtl system for axially symmetric flow. Then two types of problems for the Prandtl system with the transmission conditions on a certain surface are considered: the boundary layer near a wall through which an another fluid is injected, and the problem for two mixed fluids. After that the applications of the boundary layer theory to non-Newtonian flows and to magnetohydrodynamics are considered. By means of the method of homogenization, chapter 10 treats the problems for Prandtl system with rapidly oscillating injection and suction and the MHD problem in a rapidly oscillating magnetic field. The monograph is finished by stating some open problems in the boundary layer theory.

The level of this interesting book is very high, but clear. The exposition is comprehensive, and theorems and lemmas are thoroughly proved. It seems that the authors of this monograph are the first to undertake such a successive and systematic exposition of the main mathematical results and methods in the boundary layer theory.

Reviewer: P.B.Dubovskiĭ (Moskva)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |

35Q35 | PDEs in connection with fluid mechanics |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

76M20 | Finite difference methods applied to problems in fluid mechanics |