Scaling, self-similarity, and intermediate asymptotics.

*(English)*Zbl 0907.76002
Cambridge Texts in Applied Mathematics. 14. Cambridge: Cambridge University Press. xxii, 386 p. (1996).

In this book, Prof. G. I. Barenblatt, a recognized expert in many areas such as the theory of turbulence, blast waves, filtration, fracture mechanics and geophysical flows, describes his many-year experience in understanding complex physical processes from a unified point of view, namely by using the concept of intermediate asymptotics. Generally speaking, intermediate asymptotics describe solutions of various problems in the range where the solutions no longer depend on the fine details of initial and boundary conditions, yet the system is still far from being in an ultimate equilibrium state.

The behaviour of intermediate asymptotics explains a fundamental difference between two types of self-similar solutions observed in many problems. If the intermediate asymptotics is represented by a function that tends to a finite (different from zero) limit when approaching the self-similar state, a self-similarity of the first kind appears, where the exponents in the power-type scaling can be obtained by using the classical dimensional analysis. If, however, a finite limit does not exist, then a self-similarity of the second kind appears, where the corresponding exponents depend on fine details of the preself-similar behaviour and can be usually determined from the solution of a nonlinear eigenvalue problem. So, it became clear how anomalous transcendental dimensions appear in self-similar solutions.

The author illustrates the above ideas by many examples listed in the contents. Moreover, he emphasizes an intrinsic relation between the method of intermediate asymptotics and the renormalization group approach developed in theoretical physics, where the solution can turn out to be self-similar, and the self-similar variables can be determined as a result of invariance with respect to a transformation group. Some instructive examples, in particular the boundary layer on a flat plate, clarify this idea.

The following chapter headings reflect in detail the discussed topics: 0. Introduction; 1. Dimensions, dimensional analysis and similarity; 2. The construction of intermediate-asymptotic solutions using dimensional analysis. Self-similar solutions; 3. Self-similarities of the second kind: first examples; 4. Self-similarities of the second kind: further examples; 5. Classification of similarity rules and self-similarity solutions. A recipe for application of similarity analysis; 6. Scaling and transformation groups. Renormalization group; 7. Self-similar solutions and travelling waves; 8. Invariant solutions: asymptotic conservation laws, spectrum of eigenvalues, and stability; 9. Scaling in the deformation and fracture of solids; 10. Scaling in turbulence; 11. Scaling in geophysical fluid dynamics; 12. Scaling: miscellaneous special problems.

The behaviour of intermediate asymptotics explains a fundamental difference between two types of self-similar solutions observed in many problems. If the intermediate asymptotics is represented by a function that tends to a finite (different from zero) limit when approaching the self-similar state, a self-similarity of the first kind appears, where the exponents in the power-type scaling can be obtained by using the classical dimensional analysis. If, however, a finite limit does not exist, then a self-similarity of the second kind appears, where the corresponding exponents depend on fine details of the preself-similar behaviour and can be usually determined from the solution of a nonlinear eigenvalue problem. So, it became clear how anomalous transcendental dimensions appear in self-similar solutions.

The author illustrates the above ideas by many examples listed in the contents. Moreover, he emphasizes an intrinsic relation between the method of intermediate asymptotics and the renormalization group approach developed in theoretical physics, where the solution can turn out to be self-similar, and the self-similar variables can be determined as a result of invariance with respect to a transformation group. Some instructive examples, in particular the boundary layer on a flat plate, clarify this idea.

The following chapter headings reflect in detail the discussed topics: 0. Introduction; 1. Dimensions, dimensional analysis and similarity; 2. The construction of intermediate-asymptotic solutions using dimensional analysis. Self-similar solutions; 3. Self-similarities of the second kind: first examples; 4. Self-similarities of the second kind: further examples; 5. Classification of similarity rules and self-similarity solutions. A recipe for application of similarity analysis; 6. Scaling and transformation groups. Renormalization group; 7. Self-similar solutions and travelling waves; 8. Invariant solutions: asymptotic conservation laws, spectrum of eigenvalues, and stability; 9. Scaling in the deformation and fracture of solids; 10. Scaling in turbulence; 11. Scaling in geophysical fluid dynamics; 12. Scaling: miscellaneous special problems.

Reviewer: O.Titow (Berlin)

##### MSC:

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

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

00A73 | Dimensional analysis (MSC2010) |

81T17 | Renormalization group methods applied to problems in quantum field theory |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

92C10 | Biomechanics |