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Author: Tatsuo Nishitani
Publisher: Springer
ISBN: 3319676121
Category : Mathematics
Languages : en
Pages : 215
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Book Description
Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for differential operators with non-effectively hyperbolic double characteristics. Previously scattered over numerous different publications, the results are presented from the viewpoint that the Hamilton map and the geometry of bicharacteristics completely characterizes the well/ill-posedness of the Cauchy problem. 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 eigen values. 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 terms. If there is a non-effectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between −Pμj and Pμj , where iμj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 × 4 Jordan block, the spectral structure of FPm is insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.
Author: Tatsuo Nishitani
Publisher: Springer
ISBN: 3319676121
Category : Mathematics
Languages : en
Pages : 215
Get Book Here
Book Description
Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for differential operators with non-effectively hyperbolic double characteristics. Previously scattered over numerous different publications, the results are presented from the viewpoint that the Hamilton map and the geometry of bicharacteristics completely characterizes the well/ill-posedness of the Cauchy problem. 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 eigen values. 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 terms. If there is a non-effectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between −Pμj and Pμj , where iμj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 × 4 Jordan block, the spectral structure of FPm is insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.
Author: Universidade Federal de Pernambuco. Departamento de Matemática
Publisher:
ISBN:
Category : Differential operators
Languages : en
Pages : 44
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Book Description
Author: Tatsuo Nishitani
Publisher:
ISBN: 9784864970181
Category : Mathematics
Languages : en
Pages : 0
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Book Description
Annotation 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 said to be effectively hyperbolic or noneffectively hyperbolic. For noneffectively hyperbolic operators, it was proved in the late of 1970s 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. It has been recognized that what is crucial to the C well-posedness is not only the Hamilton map but also the behavior of orbits of the Hamilton flow near the double characteristic manifold and the Hamilton map itself is not enough to determine completely the behavior of orbits of the flow. Strikingly enough, if there is an orbit of the Hamilton flow which lands tangentially on the double characteristic manifold then the Cauchy problem is not C well posed even though the Levi condition is satisfied, only well posed in much smaller function spaces, the Gevrey class of order 1 s 5 and not well posed in the Gevrey class of order s 5. In this lecture, we provide a general picture of the Cauchy problem for noneffectively hyperbolic operators, from the view point that the Hamilton map and the geometry of orbits of the Hamilton flow completely characterizes the well/not well-posedness of the Cauchy problem, exposing well/not well-posed results of the Cauchy problem with detailed proofs. Book jacket.
Author: Zuily
Publisher: Springer Science & Business Media
ISBN: 1489966560
Category : Science
Languages : en
Pages : 184
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Book Description
Author: Ti-Jun Xiao
Publisher: Springer
ISBN: 3540494790
Category : Mathematics
Languages : en
Pages : 314
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Book Description
The main purpose of this book is to present the basic theory and some recent de velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.
Author: Daniel Jordon
Publisher:
ISBN:
Category : Cauchy problem
Languages : en
Pages : 208
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Book Description
The purpose of this thesis is to ascertain whether linear differential operators with vanishing coefficients make suitable operators for Cauchy problems. Well-posedness for linear Cauchy problems - characterized by existence, uniqueness, and continuous dependence on the initial data - depends on a ray being in the spectrum of the operator and an estimate for the resolvent operator along this ray. This was originally shown by Hille and Yosida for operators when every positive real number is in the resolvent set, and later generalized by Feller, Miyadera, and Phillips. We restrict our attention to the setting where the differential operator acts on functions that depend on a spatial variable that takes values from a bounded subset of the real line. We establish ill-posedness of the Cauchy problem by analyzing the spectral properties of the differential operator and prove the spectrum is the entire complex plane for a wide variety of differential operators with vanishing coefficients. If the differential operator is the product of a polynomial in the derivative with a scaler function that has roots of finite multiplicity, we develop simple criteria for establishing ill-posedness of the Cauchy problem. We establish point spectral results when the function is real-valued with only simple roots, in particular we show the point spectrum is the entire complex plane. Much less is known when the coefficients of the differential operator depend on time. For these non-autonomous Cauchy problems (NCPs) only sufficient conditions for well-posedness are known, with necessary conditions still lacking. In this thesis we make strides with establishing necessary spectral conditions for well-posed NCPs in the case where the family of operators is continuous.
Author: Sigeru Mizohata
Publisher: Academic Press
ISBN: 148326906X
Category : Mathematics
Languages : en
Pages : 186
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Book Description
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. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. The book then takes a look at micro-local analysis in Gevrey class, including proof and consequences of theorem 1. The manuscript examines Schrödinger type equations, as well as general view-points on evolution equations. Numerical representations and analyses are provided in the explanation of these type of equations. The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem.
Author: M. Taylor
Publisher: Springer
ISBN: 3540372660
Category : Mathematics
Languages : en
Pages : 160
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Book Description
These notes are based on the lectures given on partial differential equations at the University of Michigan during the winter semester of 1972, with some extensions. The students to whom these lectures were addressed were assumed to have knowledge of elementary functional analysis, the Fourier transform, distribution theory, and Sobolev spaces, and such tools are used without comment. In this monography, we develop one tool, the calculus of pseudo differential operators, and apply it to several of the main problems of partial differential equations.
Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 9783540225164
Category : Mathematics
Languages : en
Pages : 416
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Book Description
Author received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equations
Author: Vincenzo Ancona
Publisher: CRC Press
ISBN: 9780203911143
Category : Mathematics
Languages : en
Pages : 390
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Book Description
Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the topic, this text is key for researchers and mathematicians specializing in hyperbolic, Schrödinger, Einstein, and partial differential equations; complex analysis; and mathematical physics.
