The method of intrinsic scaling. A systematic approach to regularity for degenerate and singular PDEs.

*(English)*Zbl 1158.35003
Lecture Notes in Mathematics 1930. Berlin: Springer (ISBN 978-3-540-75931-7/pbk). x, 150 p. (2008).

This book concerns the regularity theory for degenerate and singular parabolic equations and the focus is on a particular subject – the Hölder continuity of solutions. Technique such as the method of intrinsic scaling form the cornerstone basis for the presence analysis, which is instrumental in dealing with more elaborate aspect of the theory, like the boundedness of solutions, Harnack inequalities. A classical background needed for discussions, that will serve as a prototype along the text, is the parabolic \(p\)-Laplace equation

\[ u_t - \text{div}| \nabla\,u| ^{p-2}\nabla\,u = 0, \quad p > 1, \] if \(p > 2\), the equation is degenerate at the points where \(| \nabla\,u| = 0\). If \(1 < p < 2\), the modulus of ellipticity \(| \nabla\,u| ^{p-2}\) becomes unbounded at the points where \(| \nabla\,u| \) vanishes and the equation is said to be singular. The nature and origin of the degeneracy or singularity may be quite different but it produces a common in the equation – the weakening of its structure and that some of the properties of its solutions are lost. The challenge is to understand to what extent this weakness of the structure, at the points where the equation degenerates or becomes singular, compromises the regularizing effect that is typical of parabolic equations. Unlike the elliptic case, the degeneracy or singularity of the principal part of the parabolic \(p\)-Laplace equation plays a peculiar role, and one could not establish whether a solution is locally Hölder continuous. The results on the Hölder continuity of weak solutions are paramount in this context since they assure that no singularities arise as a consequence of the degradation of the structure of the equation. These results follow from the unifying idea of intrinsic scaling: the diffusion process in the equations evolve in a time scale determining instant by instant by the solution itself, so that they can be regarded as the heat equation in their own intrinsic time-configuration. The aim of this book is to describe in details the method of intrinsic scaling for obtaining continuity results for the weak solutions of degenerate and singular parabolic equations, and to convince the reader of the strength of this approach to regularity, by giving evidence of its wide applicability in different situations.

The book consists of seven chapters which are combined in two parts. The first part of the book is essentially an edited version of the book by E. DiBenedetto, J. M. Urbano and V. Vespri [“Current issues on singular and degenerate evolution equations”, in Evolutionary equations. Vol. I. Amsterdam: Elsevier/North-Holland. Handbook of Differential Equations, 169–286 (2004; Zbl 1082.35002)]. The second part is devoted to a series of three applications of the theory to relevant models arising from flows in porous media, chemotaxis and phase transitions. In Chapter 5, the flow of two immiscible fluids through a porous medium is studied and intrinsic scaling is used to obtain the Hölder continuity of the saturation, that satisfies a partial differential equation with a two-side degeneracy. The same type of structure arises in a model of chemotaxis with volume-filling effect. In Chapter 6, the continuity of the weak solutions of the porous medium equation with variable exponent is shown, generalizing the classical result to an increasing popular context. Finally, in Chapter 7, the Stefan problem for the singular \(p\)-Laplacian is considered, and intrinsic scaling is used to derive the continuity of the temperature, showing that no jumps occur across the free boundary.

The book will be very useful for researchers from different branches of mathematical physics.

\[ u_t - \text{div}| \nabla\,u| ^{p-2}\nabla\,u = 0, \quad p > 1, \] if \(p > 2\), the equation is degenerate at the points where \(| \nabla\,u| = 0\). If \(1 < p < 2\), the modulus of ellipticity \(| \nabla\,u| ^{p-2}\) becomes unbounded at the points where \(| \nabla\,u| \) vanishes and the equation is said to be singular. The nature and origin of the degeneracy or singularity may be quite different but it produces a common in the equation – the weakening of its structure and that some of the properties of its solutions are lost. The challenge is to understand to what extent this weakness of the structure, at the points where the equation degenerates or becomes singular, compromises the regularizing effect that is typical of parabolic equations. Unlike the elliptic case, the degeneracy or singularity of the principal part of the parabolic \(p\)-Laplace equation plays a peculiar role, and one could not establish whether a solution is locally Hölder continuous. The results on the Hölder continuity of weak solutions are paramount in this context since they assure that no singularities arise as a consequence of the degradation of the structure of the equation. These results follow from the unifying idea of intrinsic scaling: the diffusion process in the equations evolve in a time scale determining instant by instant by the solution itself, so that they can be regarded as the heat equation in their own intrinsic time-configuration. The aim of this book is to describe in details the method of intrinsic scaling for obtaining continuity results for the weak solutions of degenerate and singular parabolic equations, and to convince the reader of the strength of this approach to regularity, by giving evidence of its wide applicability in different situations.

The book consists of seven chapters which are combined in two parts. The first part of the book is essentially an edited version of the book by E. DiBenedetto, J. M. Urbano and V. Vespri [“Current issues on singular and degenerate evolution equations”, in Evolutionary equations. Vol. I. Amsterdam: Elsevier/North-Holland. Handbook of Differential Equations, 169–286 (2004; Zbl 1082.35002)]. The second part is devoted to a series of three applications of the theory to relevant models arising from flows in porous media, chemotaxis and phase transitions. In Chapter 5, the flow of two immiscible fluids through a porous medium is studied and intrinsic scaling is used to obtain the Hölder continuity of the saturation, that satisfies a partial differential equation with a two-side degeneracy. The same type of structure arises in a model of chemotaxis with volume-filling effect. In Chapter 6, the continuity of the weak solutions of the porous medium equation with variable exponent is shown, generalizing the classical result to an increasing popular context. Finally, in Chapter 7, the Stefan problem for the singular \(p\)-Laplacian is considered, and intrinsic scaling is used to derive the continuity of the temperature, showing that no jumps occur across the free boundary.

The book will be very useful for researchers from different branches of mathematical physics.

Reviewer: Vladimir N. Grebenev (Novosibirsk)

##### MSC:

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

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35K65 | Degenerate parabolic equations |

35B45 | A priori estimates in context of PDEs |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |