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# Cauchy problem for hyperbolic operators multiple characteristics, micro-local approach by Karen Yagdjian

Written in English

## Subjects:

• Differential equations, Hyperbolic.,
• Partial differential operators.,
• Cauchy problem.

Edition Notes

Includes bibliographical references (p. [379]-393) and indexes.

## Book details

Classifications The Physical Object Statement Karen Yagdjian. Series Mathematical topics,, v. 12, Mathematical topics (Berlin, Germany) ;, v. 12. LC Classifications QA377 .Y34 1997 Pagination 397 p. ; Number of Pages 397 Open Library OL282719M ISBN 10 3055017390 LC Control Number 97185120 OCLC/WorldCa 37865712

A doubly characteristic point of a diﬀerential operator P of order m (i.e. one where Pm = dPm = 0) is eﬀectively hyperbolic if the Hamilton map FPm has real non-zero eigenvalues. When the characteristics are at most double and every double characteristic is eﬀectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower.

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. The Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved.

Subsequently, an existence result for the Cauchy problem is : Annamaria Barbagallo, Vincenzo Esposito. For noneffectively hyperbolic operators, it was proved in the late of s that for the Cauchy problem to be C ∞ well posed the subprincipal symbol has to be real and bounded, in modulus, by the sum of modulus of pure imaginary eigenvalues of the Hamilton map.

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 book is a valuable reference for mathematicians and researchers interested in the Cauchy problem. Show less Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems.

This book collects together detailed proofs of many well/ill-posed results related to the Cauchy problem for diﬀerential operators. When the characteristics are at most double and every double characteristic is eﬀectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order : Springer International Publishing.

The Strictly Hyperbolic Cauchy Problem. The Basic Calculus Necessary Conditions for Correctness of the Cauchy Problem. Hyperbolic Operators of Principal Type In he was awarded the Fields Medal for his contributions to the general theory of linear partial differential operators.

His book Linear Partial. necessary and sufﬁcient for the Cauchy problem to be C∞well-posed. Introduction It is well-known that the Cauchy problem for a hyperbolic operator with con-stant coefﬁcients is C∞well-posed if and only if the operator is hyperbolic in the sense of Garding (see [5]).

For hyperbolic operators with constant coefﬁ-˚. - weakly hyperbolic Cauchy problems with finite time degeneracy; the precise loss of regularity depending on the spatial variables is determined; the main step is to find the correct class of pseudodifferential symbols and to establish a calculus which contains a symmetrizer.

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 Cauchy problem for hyperbolic operators book.

This method can be applied to other (non) hyperbolic. On the Cauchy Problem for Hyperbolic Operators with Double Characteristics. Communications in Partial Differential Equations: Vol. 34, No. 8, pp. How to read online Cauchy Problem for Noneffectively Hyperbolic Operators (MSJ Memoirs) ePub books.

- At a double characteristic point of a differential operator with real characteristics, the linearization of the Hamilton vector field of the principal symbol is called the Hamilton map and according to either the Hamilton map has non zero real eigenvalues or not, the operator is.

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 very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and.

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.

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators Alessia Ascanellia, Massimo Cicognanib,c,∗ a Dipartimento di Matematica, Università di Ferrara, Via Machiave Ferrara, Italy b Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, Bologna, Italy.

2 Cauchy Problem In this section we formulate the Cauchy problem for a linear diﬀerential operator a x, ∂ ∂x!. To begin with we make a few formal reductions. Let S be a hypersurface in Rn deﬁned by an equation ϕ(x) = 0 where ϕis a suﬃciently often continuously diﬀerentiable function with its gradient ϕx(x0) ≡ ∂ϕ ∂x1 (x0.

Corpus ID: Lectures on Cauchy problem @inproceedings{LecturesOC, title={Lectures on Cauchy problem}, author={茂 溝畑 and M. Murthy and B. Singbal}, year={} }. The cauchy problem for a weakly hyperbolic equation with unbounded and non-Lipschitz-continuous coefficients Ascanelli, Alessia, Differential and Integral Equations, On the Cauchy problem for hyperbolic operators of second order whose coefficients depend only on the time variable WAKABAYASHI, Seiichiro, Journal of the Mathematical Society.

HYPERBOLIC PROBLEMS AND REGULARITY QUESTIONS, MARIAROSARIA PADULA Books, SPRINGER Books, at Meripustak. some ill-posed Cauchy problems for linear PDEs can be generated by unbounded linear operators of those PDEs.

These are those operators for which Carleman estimates are valid, e.g. elliptic, parabolic and hyperbolic operators of the second order. Convergence rates of minimizers are established using Carleman estimates.

Nishitani, The Cauchy problem for e↵ectively hyperbolic operators, in Nonlinear variational problems (Isola d'Elba, ), Res. Notes in Math. (Pitman, ), pp. Nonlinear. (weakly) hyperbolic diﬀerential operators.

As it was proved by Bony-Schapira [4] for a single diﬀerential operator, and by Kashiwara-Schapira [13] for general systems, the hyperbolic Cauchy problem is well posed in the sheaf Bof Sato’s hyperfunctions: Theorem Let f N: N →M be a morphism of real analytic manifolds, and let.

The hyperbolic Cauchy problem by Tatsuo Nishitani Department of Mathematics, College of General Education Osaka University, Toyonaka, OsakaJapan (Soeul National University, February, ) 1. Review on basic facts Hyperbolicity Let P be a diﬀerential operator of order m deﬁned on an open set Ω in IRd+1 and let H be a.

Purchase Scattering Theory for Hyperbolic Operators, Volume 21 - 1st Edition. Print Book & E-Book. ISBN hyperbolic differential operators and related problems lecture notes in pure and applied mathematics Posted By Corín TelladoMedia TEXT ID bded1e Online PDF Ebook Epub Library Derivative And Integral Of Trigonometric And Hyperbolic.

In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first − 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 study.

Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v, In this monograph we discuss the ∞ well-posedness of the Cauchy problem for hyperbolic systems.

Usage notes. The hypersurface S is called the Cauchy. We consider the well-posedness of the Cauchy problem in Gevrey spaces for N×N first-order weakly hyperbolic systems. The question is to know whether the general results of Bronštein [1 Bronštein, M.D.

().The Cauchy Problem for hyperbolic operators with characteristic of variable multiplicity. Project Euclid - mathematics and statistics online. On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness Bernardi, Enrico and Nishitani, Tatsuo, Kyoto Journal of Mathematics, ; Non-autonomous Ornstein-Uhlenbeck equations in exterior domains Hansel, Tobias and Rhandi, Abdelaziz, Advances in Differential Equations,   If the characteristic Cauchy problem has a solution, then the equation and the second initial condition imply a necessary condition for its solvability: $\nu'(x)=0$, i.e.

the characteristic Cauchy problem may be solvable only if $\nu(x)=\mathrm{const}=\alpha$. hyperbolic differential operators and related problems lecture notes in pure and applied mathematics Posted By Robert LudlumPublic Library TEXT ID bded1e Online PDF Ebook Epub Library hyperbolic differential operators and related problems by vincenzo ancona and jean vaillant abstract presenting research from more than 30 international authorities this reference provides a.

The text then ponders on the Cauchy problem for effectively hyperbolic equations and for uniformly diagonalizable hyperbolic systems in Gevrey classes.

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.

of operators for which the Cauchy problem is weakly well posed. Such results are described in and Taylor on the use of pseudodiﬀerential techniques in nonlinear problems.

Other books on hyperbolic partial diﬀerential equations include those of Hadamard, Leray, G˚arding, and Mizohata The deﬁning properties of hyperbolic problems. operators are unique and that they extend to several spaces of sections.

We argue that Green-hyperbolic operators are not necessarily hyperbolic in any PDE-sense and that they cannot be characterized in general by well-posedness of a Cauchy problem.

The fourth section is devoted to extending the Green’s operators to distributional sections. In this paper, we study the well posed‐ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane.

By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter m ∈(0,1) and. BEHAVIOR OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A HYPERBOLIC EQUATION AS t —» B. VAlNBERG UDC Introduction Suppose that P(id/dt, id/dx), χ - (χ, • •, x), is a homogeneous hyperbolic operator without multiple characteristics, with constant coefficients and of order 2m, This means that for σ.

In particular, symbolic symmetrizers are shown to exist for constantly hyperbolic operators. Subsequently, well-posedness is proved in Sobolev spaces Hs, for any s in the case of infinitely smooth coefficients, and also for s large enough in the case of Hs coefficients.

Uniqueness follows from energy estimates for the direct Cauchy problem. B 2 − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data.

An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0.About this Item: Oxford University Press, United Kingdom, Hardback.

Condition: New. New. Language: English. Brand new Book. 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.hyperbolic differential operators and related problems lecture notes in pure and applied mathematics Posted By Cao XueqinMedia TEXT ID bded1e Online PDF Ebook Epub Library Hyperbolic Trig Identities List Of Hyperbolic.

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