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7 produkter

Del 2097 - Lecture Notes in Mathematics
Hyperbolic Systems with Analytic Coefficients
Well-posedness of the Cauchy Problem
AvTatsuo Nishitani
Häftad, Engelska, 2013
553 kr
Skickas inom 10-15 vardagar
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 order term?For first order two by two systems with two independent variables with real analytic coefficients, we present complete answers for both (A) and (B). For first order systems with real analytic coefficients we prove general necessary conditions for question (B) in terms of minors of the principal symbols. With regard to sufficient conditions for (B), we introduce hyperbolic systems with nondegenerate characteristics, which contain strictly hyperbolic systems, and prove that the Cauchy problem for hyperbolic systems with nondegenerate characteristics is well posed for any lower order term. We also prove that any hyperbolic system which is close to a hyperbolic system with a nondegenerate characteristic of multiple order has a nondegenerate characteristic of the same order nearby.
Hyperbolic Systems with Analytic Coefficients
Well-posedness of the Cauchy Problem
AvTatsuo Nishitani
E-bok
PDF, Engelska, 2013
710 kr
Läs direkt efter köp
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 order term?For first order two by two systems with two independent variables with real analytic coefficients, we present complete answers for both (A) and (B). For first order systems with real analytic coefficients we prove general necessary conditions for question (B) in terms of minors of the principal symbols. With regard to sufficient conditions for (B), we introduce hyperbolic systems with nondegenerate characteristics, which contain strictly hyperbolic systems, and prove that the Cauchy problem for hyperbolic systems with nondegenerate characteristics is well posed for any lower order term. We also prove that any hyperbolic system which is close to a hyperbolic system with a nondegenerate characteristic of multiple order has a nondegenerate characteristic of the same order nearby.
Del 2202 - Lecture Notes in Mathematics
Cauchy Problem for Differential Operators with Double Characteristics
Non-Effectively Hyperbolic Characteristics
AvTatsuo Nishitani
Häftad, Engelska, 2017
551 kr
Skickas inom 10-15 vardagar
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.
Cauchy Problem for Differential Operators with Double Characteristics
Non-Effectively Hyperbolic Characteristics
AvTatsuo Nishitani
E-bok
Engelska, 2017
710 kr
Läs direkt efter köp
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.
Hyperbolic Cauchy Problem
AvTatsuo Nishitani,Kunihiko Kajitani
E-bok
PDF, Engelska, 2006
367 kr
Läs direkt efter köp
The approach to the Cauchy problem taken here by the authorsis based on theuse of Fourier integral operators with acomplex-valued phase function, which is a time functionchosen suitably according to the geometry of the multiplecharacteristics. The correctness of the Cauchy problem inthe Gevrey classes for operators with hyperbolic principalpart is shown in the first part. In the second part, thecorrectness of the Cauchy problem for effectively hyperbolicoperators is proved with a precise estimate of the loss ofderivatives. This method can be applied to other (non)hyperbolic problems. The text is based on a course oflectures given for graduate students but will be of interestto researchers interested in hyperbolic partial differentialequations. In the latter part the reader is expected to befamiliar with some theory of pseudo-differential operators.
Del 1505 - Lecture Notes in Mathematics
Hyperbolic Cauchy Problem
AvKunihiko Kajitani,Tatsuo Nishitani
Häftad, Engelska, 1991
278 kr
Skickas inom 10-15 vardagar
The approach to the Cauchy problem taken here by the authorsis based on theuse of Fourier integral operators with acomplex-valued phase function, which is a time functionchosen suitably according to the geometry of the multiplecharacteristics. The correctness of the Cauchy problem inthe Gevrey classes for operators with hyperbolic principalpart is shown in the first part. In the second part, thecorrectness of the Cauchy problem for effectively hyperbolicoperators is proved with a precise estimate of the loss ofderivatives. This method can be applied to other (non)hyperbolic problems. The text is based on a course oflectures given for graduate students but will be of interestto researchers interested in hyperbolic partial differentialequations. In the latter part the reader is expected to befamiliar with some theory of pseudo-differential operators.
Del 30 - MATHEMATICAL SOCIETY OF JAPAN MEMOIRS
CAUCHY PROBLEM FOR NONEFFECTIVELY HYPERBOLIC OPERATORS
AvTatsuo Nishitani
Häftad, Engelska, 2014
302 kr
Skickas inom 11-20 vardagar
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.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

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Tatsuo Nishitani – författare

Hyperbolic Systems with Analytic Coefficients

Hyperbolic Systems with Analytic Coefficients

Cauchy Problem for Differential Operators with Double Characteristics

Cauchy Problem for Differential Operators with Double Characteristics

Hyperbolic Cauchy Problem

Hyperbolic Cauchy Problem

CAUCHY PROBLEM FOR NONEFFECTIVELY HYPERBOLIC OPERATORS

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