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This chapter examines the Cauchy problem, or pure initial value problem, for the PME and the GPME in d-dimensional space, d = 1. It concentrates the main effort on the PME with zero forcing term. It considers solutions which are integrable with respect to the space variables, so-called solutions with finite mass, and develops the corresponding L ¹ theory. It establishes well-posedness for the Cauchy problem in this setting, which is the one most often found in relevant literature and applications.

This chapter introduces a study of the properties of the support and the free boundary of the solution in a several dimensional setting. It discusses in detail the property of finite propagation and its consequences for the PME. Attention is focused on non-negative solutions.

This chapter begins with a systematic study of the questions of existence, uniqueness, and main properties of the solutions of the PME by concentrating on the first boundary-value problem posed in a spatial domain Ω, which is a bounded subdomain of ℝ ^d , d ≥ 1. It focuses on homogeneous Dirichlet boundary conditions, u = 0 on ∂Ω, in order to obtain a simple problem for which a fairly complete theory can be easily developed as a first stage in understanding the theory of the PME. This is called the homogeneous Cauchy-Dirichlet problem, or more simply, the homogeneous Dirichlet problem.

This chapter begins with a study of the behaviour of solutions of the PME for large times. The cornerstone of the presentation is the interplay between asymptotic behaviour and self-similarity. It is also shown that large time behaviour gives rise to the formation of patterns. Section 18.2 contains a proof of the asymptotic theorem for non-negative solutions using the so-called four step method, based on rescaling and compactness. The convergence of supports and interfaces for compactly supported data occupies Section 18.3. Section 18.4 examines the so-called continuous scaling and the associated Fokker-Planck equations. Section 18.6 introduces another functional, the entropy. Section 18.7 delves in to the peculiarities of asymptotic behaviour in one space dimension; this allows us to establish optimal convergence rates. Section 18.8 contains a proof of asymptotic convergence for signed solutions, and the extension to cover integrable forcing terms. Section 18.9 gives an introduction to the special properties of the large time behaviour of the FDE.

This chapter begins with a discussion of moving frame formulae. It then covers n + 1 splitting adapted to space slices, constraints and evolution, Hamiltonian and symplectic formulation, Cauchy problem, wave gauges, local existence for the full Einstein equations, constraints in a wave gauge, and Einstein equations with field sources.

This chapter presents an introduction to some of the main topics relating to the PME, focusing on non-negative solutions. Section 15.1 presents a detailed analysis of the regularity of the pressure, for which Lipschitz continuity is proved both in space and time. Section 15.2 introduces new comparison results. Shifting comparison, intersection comparison and lap number count are quite useful in the study of interfaces. The study of interfaces is begun in Section 15.3. The growth of the interface is estimated and the waiting time analysed. Section 15.4 deals with some of the main issues of the theory.

This chapter starts with a brief historical excursus of the early methods used for the numerical solution of DDEs. It then considers general formulation and convergence results for discrete and continuous methods for ODEs, which constitute the basis for the standard approach that is extensively treated in this book for solving DDEs numerically. Although the book focuses on the class of continuous Runge-Kutta methods, more general multistep methods are considered for the sake of completeness. The main features of the standard approach for DDEs and for neutral DDEs are introduced and discussed. The classical Bellman method of steps and waveform relaxation methods are described. Another innovative approach based on the transformation of the DDE into an abstract Cauchy problem is solved by a standard ODE method.

This chapter studies the existence and uniqueness of solutions of the Cauchy problem for the PME posed in the whole space, which take a Radon measure as initial data. Section 13.1 constructs limit solutions for data measures with the growth condition found as optimal in the previous chapter (in the non-negative case). The theory is continued in Section 13.2 where it is proven that any non-negative solution defined in a domain Q_T has a unique initial trace. In Sections 13.3 and 13.4, it is proved that the initial trace determines the solution in a unique way. This is a landmark in the theory of the PME and completes the basic theory of the Cauchy problem developed in previous chapters.

While all previous chapters are devoted to linear hyperbolic theories of thermoelasticity, this one concerns a rigid but nonlinear hyperbolic heat conductor due to Coleman et al. (1982, 1983, 1986). This particular material model obeys the law of conservation of energy, the dissipation inequality, Maxwell‐Cattaneo's equation, and a generalized energy‐entropy relation with a parabolic variation of the energy and entropy along the heat flux axis. Following a review of the field equations for a 1‐D case, a number of closed‐form solutions to the nonlinear governing equations are obtained, and then a method of weakly nonlinear geometric optics is applied to obtain an asymptotic solution to the Cauchy problem with a weakly perturbed initial condition associated with the nonlinear model.

When physics tells its story of the world, it writes on spatial pages and we flip pages in the temporal directions. The present moment contains the seeds of what happens next. Relativity challenges many of our pre-theoretical thoughts about time, yet even this would-be destroyer of time adheres to the idea that production or determination runs along the set of temporal directions. We might think of this fact as one of the last remnants left of manifest time in physics. Is even this residue of manifest time safe from physics? Looking at the world sideways, can we march “initial” data from “east” to “west” as well as from earlier to later? Or put even more loosely: can physics tell its stories if we write on non-spatial pages and read in non-temporal directions?

This chapter addresses the stability analysis of numerical methods, mainly Runge-Kutta, with respect to the test equations introduced in Chapter 9. Due to the many-sided stability requirements and the large variety of different test equations, several definitions of stability are investigated. The chapter distinguishes between two concepts of stability — for all delays and for fixed delay — the latter being stronger than the former. In the corresponding analysis of Runge-Kutta methods, some severe order barriers are detected, especially with respect to stability for fixed delay. With respect to the most general linear systems, the standard approach reveals it to be unsatisfactory. Alternative approaches, based on restating the delay equation as an abstract Cauchy problem, overcome order barriers and show good potential. The chapter ends with some specific additional stability issues which have been developed in the literature or are still in progress.

The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.

This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolvent and investigating the off-diagonal behavior of the heat kernel in many variables.

This chapter proves existence of solutions to the inhomogeneous problem using the Schauder estimate and analyzes a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The discussion centers on the solution w = v + u, where v solves the homogeneous Cauchy problem with v(x, 0) = f(x) and u solves the inhomogeneous problem with u(x, 0) = 0. The chapter first provides definitions for the Wright–Fisher–Hölder spaces on a general compact manifold with corners before explaining the steps involved in the existence proof. It then verifies the induction hypothesis and treats the k = 0 case. It also shows how to perform the doubling construction for P and considers the existence of the resolvent operator and a contraction semi-group. Finally, it discusses the problem of higher regularity.

This chapter introduces the model problems and the solution operator for the associated heat equations. These operators give a good approximation for the behavior of the heat kernel in neighborhoods of different types of boundary points. The chapter states and proves the elementary features of these operators and shows that the model heat operators have an analytic continuation to the right half plane. It first considers the model problem in 1-dimension and in higher dimensions before discussing the solution to the homogeneous Cauchy problem. It then describes the first steps toward perturbation theory and constructs the solution operator for generalized Kimura diffusions on a suitable scale of Hölder spaces. It also defines the resolvent families and explains why the estimates obtained here are not adequate for the perturbation theoretic arguments needed to construct the solution operator for generalized Kimura diffusions.

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

This chapter presents the main theorem, its main conclusions, as well as a full strategy of its proof, divided in nine supporting intermediate results, Theorems M0–M8. The chapter specifies the closeness to Schwarzschild of the initial data in the context of the Characteristic Cauchy problem. The conclusions of the main theorem can be immediately extended to the case where the data are specified to be close to Schwarzschild on a spacelike hypersurface Σ‎. The chapter then outlines the main bootstrap assumptions. It also provides a short description of the results concerning the General Covariant Modulation procedure, which is at the heart of the proof.

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining first and second derivative estimates in the latter case, before discussing a general result describing the off-diagonal and long-time behavior of the solution kernel for the general model. It also states a proposition summarizing the properties of the resolvent operator as an operator on the Hölder spaces. In contrast to the case of the heat equation, there is no need to assume that the data has compact support in the x-variables to prove estimates when k > 0.

This chapter establishes Hölder space estimates for higher dimensional corner model problems. It first explains the homogeneous Cauchy problem before estimating the solution of the inhomogeneous problem in a n-dimensional corner. It then reduces the proof of an estimate in higher dimensions to the estimation of a product of 1-dimensional integrals. Using the “1-variable-at-a-time” method, the chapter proves the higher dimensional estimates in several stages by considering the “pure corner” case where m = 0, and then turns to the Euclidean case, where n = 0. It also discusses the resolvent operator as the Laplace transform of the heat kernel.