# American Institute of Mathematical Sciences

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

 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

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.
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).   Google Scholar [2] H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).   Google Scholar [3] Calderón, Commutators of singular integral operators,, Proc. Nat. Acad. Sci., 53 (1965), 1092.  doi: 10.1073/pnas.53.5.1092.  Google Scholar [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.  Google Scholar [5] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 5, 5 (1992).   Google Scholar [6] H. O. Fattorini, "The Cauchy problem,", Encyclopedia Math. Appl., 18 (1983).   Google Scholar [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).   Google Scholar [8] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation,, Nonlinear Analysis, 15 (1990), 479.   Google Scholar [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.  Google Scholar [10] T. Kato, "Perturbation Theory for Linear Operators,", Grundlehren math. Wissenschaften 132, 132 (1966).   Google Scholar [11] T. Kato, Linear evolution equations of "hyperbolic" type,, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241.   Google Scholar [12] T. Kato, Linear evolution equations of "hyperbolic" type, II,, J. Math. Soc. Japan, 25 (1973), 648.  doi: 10.2969/jmsj/02540648.  Google Scholar [13] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Lecture Notes in Math., 448 (1975), 25.   Google Scholar [14] T. Kato, Singular perturbation and semigroup theory,, in Lecture Notes in Math., 565 (1976), 104.   Google Scholar [15] T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,", Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, (1985).   Google Scholar [16] T. Kato, Abstract evolution equations, linear and quasilinear, revisited,, in Lecture Notes in Math., 1540 (1993), 103.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications,, in, 7 (1996), 303.   Google Scholar [21] N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 8 (1998), 399.   Google Scholar [22] N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 3 (): 267.   Google Scholar [23] N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space,, SUT J. Math., 29 (1993), 51.   Google Scholar [24] E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Math. Soc. Monograph, (2005).   Google Scholar [25] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Math. Sci., 44 (1983).   Google Scholar [26] H. Tanabe, "Equations of Evolution,", Monographs and Studies in Math., 6 (1979).   Google Scholar [27] H. Tanabe, "Functional Analytic Methods for Partial Differential Equations,", Pure and Applied Mathmatics, 204 (1997).   Google Scholar [28] N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families,, Math. Nachr., 274/275 (2004), 130.   Google Scholar [29] B. Thaller, "The Dirac Equation,", Texts and Monographs in Physics, (1992).   Google Scholar [30] A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces,, Osaka J. Math., 16 (1979), 301.   Google Scholar

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##### References:
 [1] H. Brézis, "Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,", Mathematics Studies 5, 5 (1973).   Google Scholar [2] H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).   Google Scholar [3] Calderón, Commutators of singular integral operators,, Proc. Nat. Acad. Sci., 53 (1965), 1092.  doi: 10.1073/pnas.53.5.1092.  Google Scholar [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.  Google Scholar [5] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 5, 5 (1992).   Google Scholar [6] H. O. Fattorini, "The Cauchy problem,", Encyclopedia Math. Appl., 18 (1983).   Google Scholar [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).   Google Scholar [8] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation,, Nonlinear Analysis, 15 (1990), 479.   Google Scholar [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.  Google Scholar [10] T. Kato, "Perturbation Theory for Linear Operators,", Grundlehren math. Wissenschaften 132, 132 (1966).   Google Scholar [11] T. Kato, Linear evolution equations of "hyperbolic" type,, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241.   Google Scholar [12] T. Kato, Linear evolution equations of "hyperbolic" type, II,, J. Math. Soc. Japan, 25 (1973), 648.  doi: 10.2969/jmsj/02540648.  Google Scholar [13] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Lecture Notes in Math., 448 (1975), 25.   Google Scholar [14] T. Kato, Singular perturbation and semigroup theory,, in Lecture Notes in Math., 565 (1976), 104.   Google Scholar [15] T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,", Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, (1985).   Google Scholar [16] T. Kato, Abstract evolution equations, linear and quasilinear, revisited,, in Lecture Notes in Math., 1540 (1993), 103.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications,, in, 7 (1996), 303.   Google Scholar [21] N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 8 (1998), 399.   Google Scholar [22] N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space,, Adv. Math. Sci. Appl., 3 (): 267.   Google Scholar [23] N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space,, SUT J. Math., 29 (1993), 51.   Google Scholar [24] E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Math. Soc. Monograph, (2005).   Google Scholar [25] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Math. Sci., 44 (1983).   Google Scholar [26] H. Tanabe, "Equations of Evolution,", Monographs and Studies in Math., 6 (1979).   Google Scholar [27] H. Tanabe, "Functional Analytic Methods for Partial Differential Equations,", Pure and Applied Mathmatics, 204 (1997).   Google Scholar [28] N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families,, Math. Nachr., 274/275 (2004), 130.   Google Scholar [29] B. Thaller, "The Dirac Equation,", Texts and Monographs in Physics, (1992).   Google Scholar [30] A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces,, Osaka J. Math., 16 (1979), 301.   Google Scholar
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