This volume is the result of a workshop, “Partial Differential Equations and Fluid Mechanics”, which took place in the Mathematics Institute at the University of Warwick, May 21st–23rd, 2007.

Several of the speakers agreed to write review papers related to their contributions to the workshop, while others have written more traditional research papers. All the papers have been carefully edited in the interests of clarity and consistency, and the research papers have been externally refereed. We are very grateful to the referees for their work. We believe that this volume therefore provides an accessible summary of a wide range of active research topics, along with some exciting new results, and we hope that it will prove a useful resource for both graduate students new to the area and to more established researchers.

We would like to express their gratitude to the following sponsors of the workshop: the London Mathematical Society, the Royal Society, via a University Research Fellowship awarded to James Robinson, the North American Fund and Research Development Fund schemes of Warwick University, and the Warwick Mathematics Department (via MIR@W). JCR is currently supported by the EPSRC, grant EP/G007470/1.

Finally it is a pleasure to thank Yvonne Collins and Hazel Higgens from the Warwick Mathematics Research Centre for their assistance during the organization of the workshop.

Abstract

This paper gives a brief summary of some of the main results concerning the regularity of solutions of the three-dimensional Navier–Stokes equations. We then outline the basis of a numerical algorithm that, at least in theory, can verify regularity for all initial conditions in any bounded subset of H1.

Introduction

The aim of this paper is to present some partial results concerning the problem of regularity of global solutions of the three-dimensional Navier–Stokes equations. Since these equations form the fundamental model of hydrodynamics it is a matter of great importance whether or not they can be uniquely solved. However, one hundred and fifty years after the Navier–Stokes model was presented for the first time, we still lack an existence and uniqueness theorem, and the most significant contributions to the subject remain those of Leray (1934) and Hopf (1951).

Nevertheless, there have been many advances since their work, and it would not be possible to give an exhaustive presentation of these in a short article. We give a brief overview of some of the main results, and then concentrate on one specific and in some ways non-standard approach to the problem, with a discussion of the feasibility of testing for regularity via numerical computations following Chernysehnko et al. (2007), Dashti & Robinson (2008), and Robinson & Sadowski (2008). In some ways this contribution can be viewed as a companion to the introductory review by Robinson (2006).

This book provides a self-contained mathematical exposition of the theory of monochromatic wave propagation in layered viscoelastic media. It provides analytic solutions and numerical results for fundamental wave-propagation problems in arbitrary linear viscoelastic media not published previously in a book. As a text book with numerical examples and problem sets, it provides the opportunity to teach the theory of monochromatic wave propagation as usually taught for elastic media in the broader context of wave propagation in any media with a linear response without undue complications in the mathematics. Formulations of the expressions for the waves and the constitutive relation for the media afford considerable generality and simplification in the mathematics required to derive analytic solutions valid for any viscoelastic solid including an elastic medium. The book is intended for the beginning student of wave propagation with prerequisites being knowledge of differential equations and complex variables.

As a reference text, this book provides the theory of monochromatic wave propagation in more than one dimension developed in the last three to four decades. As such, it provides a compendium of recent advances that show that physical characteristics of two- and three-dimensional anelastic body and surface waves are not predictable from the theory for one-dimensional waves. It provides the basis for the derivation of results beyond the scope of the present text book. The theory is of interest in the broad field of solid mechanics and of special interest in seismology, engineering, exploration geophysics, and acoustics for consideration of wave propagation in layered media with arbitrary amounts of intrinsic absorption, ranging from low-loss models of the deep Earth to moderate-loss models for soils and weathered rock.

A theoretical closed-form solution for the problem of a general P, SI, or SII wave incident on a plane welded boundary between HILV media, V and V′, is one for which the characteristics of the reflected and refracted waves are expressed in terms of the assumed characteristics of the incident wave. Application of the boundary conditions at the boundary allows the amplitude and phase for the reflected and refracted waves to be expressed in terms of the properties of the media and those given for the incident wave. The directions of the propagation and attenuation vectors for the reflected and refracted waves are determined in terms of those of the incident wave by showing that the complex wave number for each solution must be the same. For problems involving incident P and SI waves, the boundary conditions are most readily applied using the solutions involving displacement potentials, namely, (4.2.1), (4.2.2), (4.2.16), and (4.2.17). For problems involving incident SII waves, the boundary conditions can be applied most easily using solutions involving only one component of the displacement field, namely (4.2.26) and (4.2.27).

Boundary-Condition Equations for General Waves

The welded boundary between media V and V′ is specified mathematically by requiring that the stress and displacement are continuous across the boundary. For purposes of brevity, application of these boundary conditions to the general solutions specifying each type of wave as incident, reflected, or refracted allows a general set of equations to be derived from which a particular problem of interest can be solved by choosing the incident wave of interest.

Solutions for problems involving wave propagation in a semi-infinite half space are of interest for interpreting measurements of radiation fields at locations near or on the free surface. Solutions to these problems as derived for elastic media have formed the basis for the initial interpretation of seismograms and resultant inferences concerning the internal structure of the Earth.

Analytic solutions and corresponding numerical examples for problems involving general SI, P, and SII waves incident on the free surface of a viscoelastic half space are presented in this chapter (Borcherdt, 1971, 1988; Borcherdt and Glassmoyer, 1989; Borcherdt et al., 1989). Closed-form expressions for displacement and volumetric strain are included to facilitate understanding and interpretation of measurements as might be detected on seismometers and volumetric strain meters at or near the free surface of a viscoelastic half space.

The procedures to solve the reflection–refraction problems for a general SI, P, or SII wave incident on a free surface are analogous to those for the corresponding problems for a welded boundary. For brevity, many of the expressions and results in medium V for a welded boundary applicable to the free-surface problems will be referred to here, but not rewritten.

Boundary-Condition Equations

Solutions of the equations of motion for problems of general P, SI, and SII waves incident and reflected from the surface of a viscoelastic half space are specified by (4.2.1) through (4.2.45) with respect to the coordinate system illustrated in Figure (4.1.3), where medium V′ is assumed to be a vacuum.

The response of a stack of multiple layers of viscoelastic media to waves incident at the base of the stack is of special interest in seismology. Solutions of the problem for elastic media with incident homogeneous waves have proven useful in understanding the response of the Earth's crust and near-surface soil and rock layers to earthquake-induced ground shaking. Solutions are provided here for general (homogeneous or inhomogeneous) SII waves incident at the base of a stack of viscoelastic layers. The derivations of solutions for the problems of incident general P and SI waves are similar, but more cumbersome. The method for deducing solutions of the incident P and SI wave problems will be illustrated by those developed here. The results provided here for viscoelastic media include those derived for elastic media (Haskell, 1953, 1960). The method used here to derive the solutions for viscoelastic waves uses a matrix formulation similar to that initially used by Thompson (1950) and implemented with the correct boundary condition for elastic media by Haskell (1953).

To set up the mathematical framework for multilayered media consider a stack of n – 1 parallel viscoelastic layers in welded contact underlain by a viscoelastic half space. Spatial reference for the layers is provided by a rectangular coordinate system designated by (x1, x2, x3) or (x, y, z) as shown in Figure (4.1.3) with the plane x3 = z = 0 chosen to correspond to the boundary at the free surface. The layers are indexed sequentially with the index of each layer corresponding to that of its lower boundary as indicated in Figure (9.1.1).

Theoretical results in the previous chapter predict that plane harmonic waves reflected and refracted at plane anelastic boundaries are in general inhomogeneous with the degree of inhomogeneity dependent on the angle of incidence, the degree of inhomogeneity of the incident wave, and properties of the viscoelastic media. As a result physical characteristics of the waves such as phase velocity, energy velocity, phase shifts, attenuation, particle motion, fractional energy loss, direction and amplitude of maximum energy flow, and energy flow due to wave interaction vary with angle of incidence. Consequently, these physical characteristics of inhomogeneous waves propagating in a stack of anelastic layers will not be unique at each point in the stack as they are for homogeneous waves propagating in elastic media. Instead these physical characteristics of the waves will depend on the angle at which the wave entered the stack and hence the travel path of the wave through previous layers. Towards understanding the significance of these dependences of the physical characteristics on angle of incidence and inhomogeneity of the incident wave, numerical models for general SII and P waves incident on single viscoelastic boundaries are presented in this chapter. Study of this chapter, especially the first three sections, provides additional insight into the effects of a viscoelastic boundary on resultant reflected and refracted waves.

A computer code (WAVES) is used to calculate reflection–refraction coefficients and the physical characteristics of reflected and refracted general waves for the problems of general (homogeneous or inhomogeneous) plane P, SI, and SII waves incident on a plane boundary between viscoelastic media (Borcherdt et al., 1986).

Analytic closed-form solutions for problems of body- and surface-wave propagation in layered viscoelastic media will show that the waves in anelastic media are predominantly inhomogeneous with the degree of inhomogeneity dependent on angle of incidence and intrinsic absorption. Hence, it is reasonable to expect from results in the previous chapter that the physical characteristics of refracted waves in layered anelastic media also will vary with the angle of incidence and be dependent on the previous travel path of the wave. These concepts are not encountered for waves in elastic media, because the waves traveling through a stack of layers are homogeneous with their physical characteristics such as phase speed not dependent on the angle of incidence. This chapter will provide the framework and solutions for each of the waves needed to derive analytic solutions for various reflection–refraction and surface-wave problems in subsequent chapters.

Specification of Boundary

To set up the mathematical framework for considering reflection–refraction problems at a single viscoelastic boundary and surface-wave problems, consider two infinite HILV media denoted by V and V′ with a common plane boundary in welded contact (Figure (4.1.3)). For reference, the locations of the media are described by a rectangular coordinate system specified by coordinates (x1, x2, x3) or (x, y, z) with the space occupied by V described by x3 > 0, the space occupied by V′ by x3 > 0, and the plane boundary by x3 = 0. For problems involving a viscoelastic half space V′ will be considered a vacuum.