Linear evolution equations with strongly measurable families and application to the Dirac equation

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June 2011, 4(3): 723-744. doi: 10.3934/dcdss.2011.4.723

Linear evolution equations with strongly measurable families and application to the Dirac equation

Noboru Okazawa ^1, and Kentarou Yoshii ^1,

1.	Department of Mathematics, Science University of Tokyo, 1-3 Kagurazaka, Sinjuku-ku, Tokyo 162-8601, Japan, Japan

Received April 2009 Revised October 2009 Published November 2010

Abstract
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A new existence and uniqueness theorem is established for linear evolution equations of hyperbolic type with strongly measurable coefficients in a separable Hilbert space. The result is applied to the Dirac equation with time-dependent potential.

Keywords: hyperbolic type, the Dirac equation., Linear evolution equations, measurable coefficients.

Mathematics Subject Classification: Primary: 47D06, 47B44; Secondary: 35L4.

Citation: Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723

References:

[1]	H. Brézis, "Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,", Mathematics Studies 5, 5 (1973).
[2]	H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).
[3]	Calderón, Commutators of singular integral operators,, Proc. Nat. Acad. Sci., 53 (1965), 1092. doi: 10.1073/pnas.53.5.1092.
[4]	G. Da Prato and M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case,, Comm. Partial Differential Equations, 1 (1976), 585. doi: 10.1080/03605307608820022.
[5]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 5, 5 (1992).
[6]	H. O. Fattorini, "The Cauchy problem,", Encyclopedia Math. Appl., 18 (1983).
[7]	N. Hayashi, P. I. Naumkin and R. B. E. Wibowo, Nonlinear scattering for a system of nonlinear Klein-Gordon equations,, J. Math. Phys., 49 (2008).
[8]	R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation,, Nonlinear Analysis, 15 (1990), 479.
[9]	S. Ishii, Linear evolution equations $du/dt+A(t)u=0$: A case where $A(t)$ is strongly uniform-measurable,, J. Math. Soc. Japan, 34 (1982), 413. doi: 10.2969/jmsj/03430413.
[10]	T. Kato, "Perturbation Theory for Linear Operators,", Grundlehren math. Wissenschaften 132, 132 (1966).
[11]	T. Kato, Linear evolution equations of "hyperbolic" type,, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241.
[12]	T. Kato, Linear evolution equations of "hyperbolic" type, II,, J. Math. Soc. Japan, 25 (1973), 648. doi: 10.2969/jmsj/02540648.
[13]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Lecture Notes in Math., 448 (1975), 25.
[14]	T. Kato, Singular perturbation and semigroup theory,, in Lecture Notes in Math., 565 (1976), 104.
[15]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,", Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, (1985).
[16]	T. Kato, Abstract evolution equations, linear and quasilinear, revisited,, in Lecture Notes in Math., 1540 (1993), 103.
[17]	K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type,, J. Math. Soc. Japan, 31 (1979), 647. doi: 10.2969/jmsj/03140647.
[18]	N. Okazawa, Remarks on linear $m$-accretive operators in a Hilbert space,, J. Math. Soc. Japan, 27 (1975), 160. doi: 10.2969/jmsj/02710160.
[19]	N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces,, J. Math. Soc. Japan, 34 (1982), 677. doi: 10.2969/jmsj/03440677.
[20]	N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications,, in, 7 (1996), 303.
[21]	N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 8 (1998), 399.
[22]	N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 3 (): 267.
[23]	N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space,, SUT J. Math., 29 (1993), 51.
[24]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Math. Soc. Monograph, (2005).
[25]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Math. Sci., 44 (1983).
[26]	H. Tanabe, "Equations of Evolution,", Monographs and Studies in Math., 6 (1979).
[27]	H. Tanabe, "Functional Analytic Methods for Partial Differential Equations,", Pure and Applied Mathmatics, 204 (1997).
[28]	N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families,, Math. Nachr., 274/275 (2004), 130.
[29]	B. Thaller, "The Dirac Equation,", Texts and Monographs in Physics, (1992).
[30]	A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces,, Osaka J. Math., 16 (1979), 301.

show all references

References:

[1]	H. Brézis, "Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,", Mathematics Studies 5, 5 (1973).
[2]	H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).
[3]	Calderón, Commutators of singular integral operators,, Proc. Nat. Acad. Sci., 53 (1965), 1092. doi: 10.1073/pnas.53.5.1092.
[4]	G. Da Prato and M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case,, Comm. Partial Differential Equations, 1 (1976), 585. doi: 10.1080/03605307608820022.
[5]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 5, 5 (1992).
[6]	H. O. Fattorini, "The Cauchy problem,", Encyclopedia Math. Appl., 18 (1983).
[7]	N. Hayashi, P. I. Naumkin and R. B. E. Wibowo, Nonlinear scattering for a system of nonlinear Klein-Gordon equations,, J. Math. Phys., 49 (2008).
[8]	R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation,, Nonlinear Analysis, 15 (1990), 479.
[9]	S. Ishii, Linear evolution equations $du/dt+A(t)u=0$: A case where $A(t)$ is strongly uniform-measurable,, J. Math. Soc. Japan, 34 (1982), 413. doi: 10.2969/jmsj/03430413.
[10]	T. Kato, "Perturbation Theory for Linear Operators,", Grundlehren math. Wissenschaften 132, 132 (1966).
[11]	T. Kato, Linear evolution equations of "hyperbolic" type,, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241.
[12]	T. Kato, Linear evolution equations of "hyperbolic" type, II,, J. Math. Soc. Japan, 25 (1973), 648. doi: 10.2969/jmsj/02540648.
[13]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Lecture Notes in Math., 448 (1975), 25.
[14]	T. Kato, Singular perturbation and semigroup theory,, in Lecture Notes in Math., 565 (1976), 104.
[15]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,", Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, (1985).
[16]	T. Kato, Abstract evolution equations, linear and quasilinear, revisited,, in Lecture Notes in Math., 1540 (1993), 103.
[17]	K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type,, J. Math. Soc. Japan, 31 (1979), 647. doi: 10.2969/jmsj/03140647.
[18]	N. Okazawa, Remarks on linear $m$-accretive operators in a Hilbert space,, J. Math. Soc. Japan, 27 (1975), 160. doi: 10.2969/jmsj/02710160.
[19]	N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces,, J. Math. Soc. Japan, 34 (1982), 677. doi: 10.2969/jmsj/03440677.
[20]	N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications,, in, 7 (1996), 303.
[21]	N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 8 (1998), 399.
[22]	N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 3 (): 267.
[23]	N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space,, SUT J. Math., 29 (1993), 51.
[24]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Math. Soc. Monograph, (2005).
[25]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Math. Sci., 44 (1983).
[26]	H. Tanabe, "Equations of Evolution,", Monographs and Studies in Math., 6 (1979).
[27]	H. Tanabe, "Functional Analytic Methods for Partial Differential Equations,", Pure and Applied Mathmatics, 204 (1997).
[28]	N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families,, Math. Nachr., 274/275 (2004), 130.
[29]	B. Thaller, "The Dirac Equation,", Texts and Monographs in Physics, (1992).
[30]	A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces,, Osaka J. Math., 16 (1979), 301.

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