14 well posed problems the set of functions used to

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1.4Well Posed ProblemsThe set of functions used to formulate a PDE, which might include coefficients orterms in the equation itself as well as boundary and initial conditions, is collectivelyreferred to as the inputdata. The most basic question for any PDE is whether asolution exists for a given set of data. However, for most purposes we want to requiresomething more. A PDE problem is said to bewell posedif, for a given set of data:1. A solution exists.2. The solution is uniquely determined by the data.3. The solution depends continuously on the data.These criteria were formulated by Jacques Hadamard in 1902. The first two propertieshold for ODE under rather general assumptions, but not necessarily for PDE. It iseasy to find nonlinear equations that admit no solutions, and even in the linear casethere is no guarantee.The third condition, continuous dependence on the input data, is sometimes calledstability. One practical justification for this requirement is it is not possible to specifyinput data with absolute accuracy. Stability implies that the effects of small variationsin the data can be controlled.For certain PDE, especially the classical linear cases, we have a good under-standing of the requirements for well-posedness. For other important problems, for
61IntroductionFig. 1.3Numerical simulations of blood flow in the aorta. Courtesy of D. Gupta, Emory UniversityHospital, and T. Passerini, M. Piccinelli and A. Veneziani, Emory Mathematics and ComputerScienceexample in fluid mechanics, well-posedness remains a difficult unsolved conjecture.Furthermore, many interesting problems are known not to be well posed. For exam-ple, problems in image processing are frequently ill posed, because information islost due to noise or technological limitations.1.5ApproachesWe can organize the methods for handling PDE problems according to three basicgoals:1.Solving: finding explicit formulas for solutions.2.Analysis: understanding general properties of solutions.3.Approximation: calculating solutions numerically.Solving PDE is certainly worth understanding in those special cases where it ispossible. The solution formulas available for certain classical PDE provide insightthat is important to the development of the theory.The goals of theoretical analysis of PDE are extremely broad. We wish to learnas much as we can about the qualitative and quantitative properties of solutions andtheir relationship to the input data.Finally, numerical computation is the primary means by which applications ofPDE are carried out. Computational methods rely on a foundation of theoreticalanalysis, but also bring up new considerations such as efficiency of calculation.Example 1.1Figure1.3shows a set of numerical simulations modeling the insertionin the aorta of a pipe-like device designed to improve blood flow. The leftmost frameshows the aorta before surgery, and the three panes on the right model the insertion

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Term
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Derivative, The Land, Partial differential equation, David Borthwick, Universitext

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