##
**Initial-boundary value problems and the Navier-Stokes equations.**
*(English)*
Zbl 0689.35001

This book is aimed to fill a gap between elementary and rather abstract monographs on this topic under the view of computational mathematics, i.e., existence, smoothness and the admissibility of boundary conditions are fundamental among the questions that are discussed in the work. The compressible and incompressible Navier-Stokes equations were selected to illustrate the rich variety of results due to the fact that the study of the above mentioned three problems with respect to them requires the whole power of the mathematical theory that is available.

The following example may demonstrate the spirit in which the theory is developed, i.e., the strong influence of needs and particularities due to the computation of approximations to the solutions of evolutionary partial differential equations on the presentation of the material. Using difference approximations for linear problems and linearization for nonlinear problems it is shown that there is a set of \(C^{\infty}\)-data which is dense in \(L^ 2\) such that there exist related \(C^{\infty}\)- solutions. For these solutions and their derivatives estimates may be derived in terms of the data. For less smooth data it is now possible to define weak solutions over a closure argument. This manner of proceeding is much closer to the computing process than the frequently used one consisting of the definition and subsequent study of smoothness of weak solutions: If one has to compute solutions to discontinuous data then the best results are obtained if the data are approximated by smooth ones and the respective smooth solutions are calculated.

The development of the theory starts with constant coefficient Cauchy problems, a key role plays a detailed discussion of concepts of well posedness. Under the examples there are general parabolic and strongly hyperbolic systems as well as the linearized compressible Euler and Navier-Stokes equations. In the next step the Cauchy problem for linear variable coefficient second order parabolic and first order strongly hyperbolic systems in one dimension is treated. Examples include the linearized Korteweg-de Vries equation and a Cauchy problem of Schrödinger type.

A first impression of the nonlinear theory is given by a study of the Cauchy problem for the viscous and the inviscid Burgers’ equations. They serve as an example to introduce certain techniques for the investigation of more general evolutionary systems performed in the subsequent chapters. Main emphasis lies on the following four aspects: 1. local (in time) existence by linear iteration, 2. local existence and global (in time) a priori estimates provide global existence, 3. smoothing properties for the parabolic case, i.e., viscous models, 4. breakdown of smooth solutions in finite time for the hyperbolic case, i.e., inviscid models. The principles and techniques obtained here are immediately carried over to nonlinear systems in one spaces dimension.

Next topic is the Cauchy problem for systems in higher space dimensions. For the linear case most results may be generalized from those obtained for one space dimension. In similar manner short-time existence for nonlinear systems is basically achieved as in the one dimensional case. A global existence result is presented for a specific 2D parabolic system.

The energy method and the Laplace transform play an important role in the parabolic case which is discussed in detail. The key to hyperbolic problems is provided by the method of characteristics. These considerations in the one dimensional case are then combined with the techniques and results for the higher dimensional periodic Cauchy problem achieved in the previous chapters. Result is the desired existence, uniqueness and smoothness theory for initial boundary value roblems for linear second order strongly parabolic systems as well as for linear first order symmetric hyperbolic systems on specific domains in higher dimensional spaces. The linearized Euler equations serve as an example for the latter results. For cases where the energy method does not provide a sufficient condition for well posedness of hyperbolic systems the Laplace transform is introduced as an alternative method. Remarks on the generalization of such existence results to nonlinear problems conclude this part of the presentation.

The final two chapters are devoted to the application of the general theory to the incompressible Navier-Stokes equations in the spatially periodic case as well as under initial and boundary conditions.

At a first glance this monograph might seem to be just another one under many ones existing in the field. Looking more closely into it the very natural and in some sense canonical choice and organization of the material, the spirit of the presentation and the rich variety of good and illustrating examples make the study of this well readable book to an interesting and inspiring experience.

The following example may demonstrate the spirit in which the theory is developed, i.e., the strong influence of needs and particularities due to the computation of approximations to the solutions of evolutionary partial differential equations on the presentation of the material. Using difference approximations for linear problems and linearization for nonlinear problems it is shown that there is a set of \(C^{\infty}\)-data which is dense in \(L^ 2\) such that there exist related \(C^{\infty}\)- solutions. For these solutions and their derivatives estimates may be derived in terms of the data. For less smooth data it is now possible to define weak solutions over a closure argument. This manner of proceeding is much closer to the computing process than the frequently used one consisting of the definition and subsequent study of smoothness of weak solutions: If one has to compute solutions to discontinuous data then the best results are obtained if the data are approximated by smooth ones and the respective smooth solutions are calculated.

The development of the theory starts with constant coefficient Cauchy problems, a key role plays a detailed discussion of concepts of well posedness. Under the examples there are general parabolic and strongly hyperbolic systems as well as the linearized compressible Euler and Navier-Stokes equations. In the next step the Cauchy problem for linear variable coefficient second order parabolic and first order strongly hyperbolic systems in one dimension is treated. Examples include the linearized Korteweg-de Vries equation and a Cauchy problem of Schrödinger type.

A first impression of the nonlinear theory is given by a study of the Cauchy problem for the viscous and the inviscid Burgers’ equations. They serve as an example to introduce certain techniques for the investigation of more general evolutionary systems performed in the subsequent chapters. Main emphasis lies on the following four aspects: 1. local (in time) existence by linear iteration, 2. local existence and global (in time) a priori estimates provide global existence, 3. smoothing properties for the parabolic case, i.e., viscous models, 4. breakdown of smooth solutions in finite time for the hyperbolic case, i.e., inviscid models. The principles and techniques obtained here are immediately carried over to nonlinear systems in one spaces dimension.

Next topic is the Cauchy problem for systems in higher space dimensions. For the linear case most results may be generalized from those obtained for one space dimension. In similar manner short-time existence for nonlinear systems is basically achieved as in the one dimensional case. A global existence result is presented for a specific 2D parabolic system.

The energy method and the Laplace transform play an important role in the parabolic case which is discussed in detail. The key to hyperbolic problems is provided by the method of characteristics. These considerations in the one dimensional case are then combined with the techniques and results for the higher dimensional periodic Cauchy problem achieved in the previous chapters. Result is the desired existence, uniqueness and smoothness theory for initial boundary value roblems for linear second order strongly parabolic systems as well as for linear first order symmetric hyperbolic systems on specific domains in higher dimensional spaces. The linearized Euler equations serve as an example for the latter results. For cases where the energy method does not provide a sufficient condition for well posedness of hyperbolic systems the Laplace transform is introduced as an alternative method. Remarks on the generalization of such existence results to nonlinear problems conclude this part of the presentation.

The final two chapters are devoted to the application of the general theory to the incompressible Navier-Stokes equations in the spatially periodic case as well as under initial and boundary conditions.

At a first glance this monograph might seem to be just another one under many ones existing in the field. Looking more closely into it the very natural and in some sense canonical choice and organization of the material, the spirit of the presentation and the rich variety of good and illustrating examples make the study of this well readable book to an interesting and inspiring experience.

Reviewer: H.Jeggle

### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35Q30 | Navier-Stokes equations |

35K45 | Initial value problems for second-order parabolic systems |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35B65 | Smoothness and regularity of solutions to PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35K15 | Initial value problems for second-order parabolic equations |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

35K55 | Nonlinear parabolic equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

35L45 | Initial value problems for first-order hyperbolic systems |

35L50 | Initial-boundary value problems for first-order hyperbolic systems |

35L60 | First-order nonlinear hyperbolic equations |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

35A35 | Theoretical approximation in context of PDEs |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

35B45 | A priori estimates in context of PDEs |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |