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 and 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, North-Holland, Amsterdam, 1973.
[2]	H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, Paris, 1983.
[3]	Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci., U.S.A., 53 (1965), 1092-1099. 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), 585-608. doi: 10.1080/03605307608820022.
[5]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 5, Evolution Problem I, Springer-Verlag, Berlin and New York, 1992.
[6]	H. O. Fattorini, "The Cauchy problem," Encyclopedia Math. Appl., 18, Addison-Wesley, Reading, MA, 1983; Cambridge Univ. Press, New York, 1984.
[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), 103501, 24 pp.
[8]	R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation, Nonlinear Analysis, TMA, 15 (1990), 479-495.
[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), 413-424. doi: 10.2969/jmsj/03430413.
[10]	T. Kato, "Perturbation Theory for Linear Operators," Grundlehren math. Wissenschaften 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[11]	T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, Sec. I., 17 (1970), 241-258.
[12]	T. Kato, Linear evolution equations of "hyperbolic" type, II, J. Math. Soc. Japan, 25 (1973), 648-666. 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, Springer-Verlag, Berlin and New York, 1975, 25-70.
[14]	T. Kato, Singular perturbation and semigroup theory, in Lecture Notes in Math., 565, Springer-Verlag, Berlin and New York, 1976, 104-112.
[15]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems," Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, Pisa; Accademia Nazionale dei Lincei, Rome, 1985.
[16]	T. Kato, Abstract evolution equations, linear and quasilinear, revisited, in Lecture Notes in Math., 1540, Springer-Verlag, Berlin and New York, 1993, 103-125.
[17]	K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654. doi: 10.2969/jmsj/03140647.
[18]	N. Okazawa, Remarks on linear $m$-accretive operators in a Hilbert space, J. Math. Soc. Japan, 27 (1975), 160-165. 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), 677-701. doi: 10.2969/jmsj/03440677.
[20]	N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications, in "Nonlinear Analysis and Applications" (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., 7, Gakkōtosho, Tokyo, (1996), 303-317.
[21]	N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 8 (1998), 399-423.
[22]	N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 3 (1993/94), 267-283.
[23]	N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space, SUT J. Math., 29 (1993), 51-70.
[24]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monograph, Princeton Univ. Press, Princeton and Oxford, 2005.
[25]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sci., 44, Springer-Verlag, Berlin and New York, 1983.
[26]	H. Tanabe, "Equations of Evolution," Monographs and Studies in Math., 6, Pitman, London, 1979.
[27]	H. Tanabe, "Functional Analytic Methods for Partial Differential Equations," Pure and Applied Mathmatics, 204, Marcel Dekker, New York, 1997.
[28]	N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families, Math. Nachr., 274/275 (2004), 130-153.
[29]	B. Thaller, "The Dirac Equation," Texts and Monographs in Physics, Springer-Verlag, Berlin and New York, 1992.
[30]	A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces, Osaka J. Math., 16 (1979), 301-315.

show all references

References:

[1]	H. Brézis, "Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert," Mathematics Studies 5, North-Holland, Amsterdam, 1973.
[2]	H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, Paris, 1983.
[3]	Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci., U.S.A., 53 (1965), 1092-1099. 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), 585-608. doi: 10.1080/03605307608820022.
[5]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 5, Evolution Problem I, Springer-Verlag, Berlin and New York, 1992.
[6]	H. O. Fattorini, "The Cauchy problem," Encyclopedia Math. Appl., 18, Addison-Wesley, Reading, MA, 1983; Cambridge Univ. Press, New York, 1984.
[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), 103501, 24 pp.
[8]	R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation, Nonlinear Analysis, TMA, 15 (1990), 479-495.
[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), 413-424. doi: 10.2969/jmsj/03430413.
[10]	T. Kato, "Perturbation Theory for Linear Operators," Grundlehren math. Wissenschaften 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[11]	T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, Sec. I., 17 (1970), 241-258.
[12]	T. Kato, Linear evolution equations of "hyperbolic" type, II, J. Math. Soc. Japan, 25 (1973), 648-666. 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, Springer-Verlag, Berlin and New York, 1975, 25-70.
[14]	T. Kato, Singular perturbation and semigroup theory, in Lecture Notes in Math., 565, Springer-Verlag, Berlin and New York, 1976, 104-112.
[15]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems," Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, Pisa; Accademia Nazionale dei Lincei, Rome, 1985.
[16]	T. Kato, Abstract evolution equations, linear and quasilinear, revisited, in Lecture Notes in Math., 1540, Springer-Verlag, Berlin and New York, 1993, 103-125.
[17]	K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654. doi: 10.2969/jmsj/03140647.
[18]	N. Okazawa, Remarks on linear $m$-accretive operators in a Hilbert space, J. Math. Soc. Japan, 27 (1975), 160-165. 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), 677-701. doi: 10.2969/jmsj/03440677.
[20]	N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications, in "Nonlinear Analysis and Applications" (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., 7, Gakkōtosho, Tokyo, (1996), 303-317.
[21]	N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 8 (1998), 399-423.
[22]	N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 3 (1993/94), 267-283.
[23]	N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space, SUT J. Math., 29 (1993), 51-70.
[24]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monograph, Princeton Univ. Press, Princeton and Oxford, 2005.
[25]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sci., 44, Springer-Verlag, Berlin and New York, 1983.
[26]	H. Tanabe, "Equations of Evolution," Monographs and Studies in Math., 6, Pitman, London, 1979.
[27]	H. Tanabe, "Functional Analytic Methods for Partial Differential Equations," Pure and Applied Mathmatics, 204, Marcel Dekker, New York, 1997.
[28]	N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families, Math. Nachr., 274/275 (2004), 130-153.
[29]	B. Thaller, "The Dirac Equation," Texts and Monographs in Physics, Springer-Verlag, Berlin and New York, 1992.
[30]	A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces, Osaka J. Math., 16 (1979), 301-315.

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