American Institute of Mathematical Sciences

Advanced
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

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).   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

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).   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

[1]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[2]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73
[3]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039
[4]

Maria J. Esteban, Eric Séré. An overview on linear and nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 381-397. doi: 10.3934/dcds.2002.8.381
[5]

Ugur G. Abdulla. Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3379-3397. doi: 10.3934/dcds.2012.32.3379
[6]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[7]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[8]

Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
[9]

Pascal Cherrier, Albert Milani. Hyperbolic equations of Von Karman type. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 125-137. doi: 10.3934/dcdss.2016.9.125
[10]

Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020081
[11]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
[12]

Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002
[13]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
[14]

Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030
[15]

N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073
[16]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[17]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[18]

Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959
[19]

Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
[20]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences