Last edited by Jugrel
Friday, July 31, 2020 | History

2 edition of Cauchy"s problem for hyperbolic equations. found in the catalog.

Cauchy"s problem for hyperbolic equations.

Lars Ga rding

# Cauchy"s problem for hyperbolic equations.

## by Lars Ga rding

Written in English

Subjects:
• Exponential functions.

• Edition Notes

Lecture notes.

The Physical Object ID Numbers Contributions University of Chicago. Dept. of Mathematics. Pagination 151 leaves. Number of Pages 151 Open Library OL14135435M

For the Cauchy problem with smooth data, this construction can be found in the book of Courant and Hilbert , a formulation for solutions in L 1 and L ∞ is stated in . In the case of. On the cauchy problem for weakly hyperbolic equations On the cauchy problem for weakly hyperbolic equations Oleinik, O. A. 0.A. OLEINIK Moscow University We consider the Cauchy problem in the domain G{O for the equation 5 T, x c R"} (1) utt - (a"(t, X)U&, + b*(t, X)USI + bO(t, X)Ut + c (4 x)u = A t, x) with the initial conditions and aii(t, x) l i E, 2 0 in G for all E.

at a given point. A partial differential equation for which the Cauchy problem is uniquely solvable for initial data specified in a neighbourhood of on any non-characteristic surface (cf. Characteristic surface).In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. In this paper, we prove that for non-effectively hyperbolic operators with smooth double characteristics exhibiting a Jordan block of size 4 on the double manifold, the Cauchy problem is well.

Let us consider the Cauchy problem for a second-order hyperbolic equation: u¨ − n i,j 1 a ij t u x ix j n j 1 b j t u x j c t u 0, t,x ∈ 0,T ×Rn, u 0,x u 0 x,u t 0,x u 1 x,x∈Rn, where the matrix a ij t is real and symmetric for all t∈ 0,T,¨u u tt. Suppose that is strictly hyperbolic, that is, there exists λ 0 >0 such.   Kruzhkov S N The Cauchy problem in the large for non-linear equations and for certain first order quasilinear systems with several variables Dokl. Akad. Nauk SSSR MathSciNet Kruzhkov S N Soviet Math. Dokl. 5

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### Cauchy"s problem for hyperbolic equations by Lars Ga rding Download PDF EPUB FB2

Cauchy's Problem For Hyperbolic Equations: Winter And Spring Quarters,University Of Chicago [Lars Garding, G. Bergendahl] on *FREE* shipping on qualifying offers. Investigations In The Theory Of Partial Differential Equations, Technical Report, No.

Department Of Army Project. Cauchy's problem for hyperbolic equations. [Chicago, Ill., University of Chicago] (OCoLC) Document Type: Book: All Authors / Contributors: Lars Gårding; University of Chicago. Department of Mathematics. The goal of this book is a construction of the fundamental solution to the Cauchy problem for hyperbolic operators with multiple characteristics.

Well-posedness of the problem in various functional spaces as well as a propagation of singularities of the solutions are investigated, by: The correctness of the Cauchy problem in the Gevrey classes for operators with hyperbolic principal part is shown in the first part.

In the second part, the correctness of the Cauchy problem for effectively hyperbolic operators is proved with a precise estimate of the loss of derivatives. Publisher Summary. This chapter discusses Lax–Mizohata Theorem. According to this theorem, if the Cauchy problem is locally uniquely solvable at the origin, then for all ξ ɛ, all the characteristic roots at the origin λ i (0,0; ξ) should be real.

In order that the forward Cauchy problem for differential equation be uniformly H ∞-wellrosed in t ɛ [0, T], it is necessary and sufficient. In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial Cauchys problem for hyperbolic equations.

book problem for the first n − 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic of the equations of mechanics are hyperbolic, and so the. We study the Dirichlet–Cauchy problem for nonlinear hyperbolic equation of second order in a domain with edges.

The aim of this paper is to prove the regularity of. This monograph focuses on the well-posedness of the Cauchy problem for linear hyperbolic systems with matrix coefficients. Mainly two questions are discussed:(A) Under which conditions on lower order terms is the Cauchy problem well posed?(B) When is the Cauchy problem well posed for any lower.

Lectures on Cauchy Problem By Sigeru Mizohata Notes by M.K. Venkatesha Murthy and B.V. Singbal No part of this book may be reproduced in any form by print, microﬁlm or any other means with-out written permission from the Tata Institute of 3 Cauchy problem for a single equation.

Project Euclid - mathematics and statistics online. This paper is devoted to the numerical analysis of inverse problems for abstract hyperbolic differential equations. Qualitative behavior. Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of from the conditions of the Cauchy problem.

Since characteristic curves are the only curves along which solutions to partial differential equations with smooth parameters can have discontinuous derivatives, solutions to elliptic.

Ivrii-V.M. Petkov, Necessary conditions for the Cauchy problem for non strictly hyperbolic equations to be well posed, Uspehi Mat. Nauk, 29(), 3– (Russian Math. Surveys, 29(),1–70). MathSciNet zbMATH Google Scholar. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties.

Cauchy problems for hyperbolic equations and systems with variable coefficients have been studied in classical works of Petrovskii, Leret, Courant, Gording. The book takes a look at generalized Hamilton flows and singularities of solutions of the hyperbolic Cauchy problem and analytic and Gevrey well-posedness of the Cauchy problem for second order weakly hyperbolic equations with coefficients irregular in time.

The selection is a dependable reference for researchers interested in hyperbolic equations. The conference in honor of Jean Vaillant 2. Hyperbolic systems with nondegenerate characteristics 3. The Cauchy problem for hyperbolic operators dominated by the time function 4.

A remark on the Cauchy problem for a model hyperbolic operator 5. Gevrey well-posedness of the Cauchy problem. Would well repay study by most theoretical physicists."--Physics Today "An overwhelming influence on subsequent work on the wave equation."--Science Progress "One of the classical treatises on hyperbolic equations."--Royal Naval Scientific Service Delivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation.

Fourier integral operators with complex-valued phase function and the Cauchy problem for hyperbolic operators.- The effectively hyperbolic Cauchy problem. Series Title: Lecture notes in mathematics (Springer-Verlag), Responsibility: Kunihiko Kajitani, Tatsuo Nishitani.

More information: Inhaltstext. This book has been cited by the following publications. Chapter 6 - Cauchy and Goursat problems for a second-order linear hyperbolic equation with two independent variables. Riemann's method pp Chapter 13 - Cauchy problem for heat-conduction equation pp Get access.

We consider hyperbolic equations with anisotropic elliptic part and some non-Lipschitz coefficients. We prove well-posedness of the corresponding Cauchy problem in some functional spaces.

These functional spaces have finite smoothness with respect to variables corresponding to regular coefficients and infinite smoothness with respect to variables corresponding to singular coefficients. The approach to the Cauchy problem taken here by the authors is based on theuse of Fourier integral operators with a complex-valued phase function, which is a time function chosen suitably according to the geometry of the multiple characteristics.

The correctness of the Cauchy problem in the. This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves.

This area has experienced substantial progress in veryrecent years thanks to the introduction of new techniques, in particular the front tracking algorithm and.Nonlinear equations and quasi-linear systems are considered, taking into account waves and wave propagation, first-order quasi-linear systems and higher order equations, the matrix formulation of systems, the characteristics and the Cauchy problem for a general nonlinear first-order equation, and the eikonal equation.

Hyperbolic systems and their characteristics are examined, giving attention.coefficients, and the equation operator is composed of first-order linear operators. Keywords: Cauchy problem, analytic solution, fourth-order hyperbolic equations, nonstrictly hyperbolic equations.

Received: 18 April Revised: 5 May 1. Introduction.