Small viscosity and boundary layer methods. Theory, stability analysis, and applications.

*(English)*Zbl 1133.35001
Modeling and Simulation in Science, Engineering and Technology. Boston, MA: Birkhäuser (ISBN 0-8176-3390-1/hbk). xii, 194 p. (2004).

Publisher’s description: This book is an introduction to the stability analysis of noncharacteristic boundary layers, emphasizing selected topics and developing mathematical tools relevant to the study of multidimensional problems. Boundary layers are present in problems from physics, engineering, mechanics, and fluid mechanics and typically appear for problems with small diffusion. Boundary layers also occur in free boundary value problems, particularly in the analysis of shock waves.

The main goal of this presentation is to provide basic tools for the understanding of multidimensional boundary layers for systems. Included are self-contained introductions to different topics such as hyperbolic boundary value problems, parabolic systems, BKW methods, construction of profiles, introduction to the theory of Evans’ functions, and energy methods with Kreiss’ symmetrizers.

Part I is devoted to linear and semilinear problems. For simplicity, the analysis restricts its attention to constant coefficients of systemic dissipative systems. An important feature of this section is the derivation of energy estimates independent of viscosity.

Part II is a treatment of quasilinear problems; the equation that governs the rapid variation inside the layer is derived and subsequently studied, allowing for the examination of multidimensional stability of planar layers.

This monograph is a valuable text for researchers, practitioners, and graduate students in applied mathematics, mathematical physics, and engineering and will be a useful supplement for the study of mathematical models in the applied sciences. Prerequisites for the reader include standard courses in analysis, integration theory, and PDEs.

The main goal of this presentation is to provide basic tools for the understanding of multidimensional boundary layers for systems. Included are self-contained introductions to different topics such as hyperbolic boundary value problems, parabolic systems, BKW methods, construction of profiles, introduction to the theory of Evans’ functions, and energy methods with Kreiss’ symmetrizers.

Part I is devoted to linear and semilinear problems. For simplicity, the analysis restricts its attention to constant coefficients of systemic dissipative systems. An important feature of this section is the derivation of energy estimates independent of viscosity.

Part II is a treatment of quasilinear problems; the equation that governs the rapid variation inside the layer is derived and subsequently studied, allowing for the examination of multidimensional stability of planar layers.

This monograph is a valuable text for researchers, practitioners, and graduate students in applied mathematics, mathematical physics, and engineering and will be a useful supplement for the study of mathematical models in the applied sciences. Prerequisites for the reader include standard courses in analysis, integration theory, and PDEs.

##### MSC:

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

35B25 | Singular perturbations in context of PDEs |

35K65 | Degenerate parabolic equations |

35L60 | First-order nonlinear hyperbolic equations |

35L67 | Shocks and singularities for hyperbolic equations |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |