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MACHIHARA Shuji
| Mathematics, Electronics and Informatics Division | Professor |
| Mathematics |
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■ MISC- Scattering theory for the Dirac equation with a non-local term
Shuji Machihara; Kimitoshi Tsutaya
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, Volume：139, Number：4, First page：867, Last page：878, 2009
Consider a scattering problem for the Dirac equation with a non-local term including the Hartree type, say the cubic convolution term. We show the existence of scattering operators for small initial data in the subcritical and critical Sobolev spaces.
ROYAL SOC EDINBURGH, English
DOI：https://doi.org/10.1017/S0308210507000479
DOI ID：10.1017/S0308210507000479, ISSN：0308-2105, eISSN：1473-7124, Web of Science ID：WOS:000268561500012 - The Cauchy problem for the 1-D Dirac-Klein-Gordon equation
Shuji Machihara
Nonlinear Differential Equations and Applications, Volume：14, Number：5-6, First page：625, Last page：641, Dec. 2007
The Cauchy problem for the Dirac-Klein-Gordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein-Gordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with. © 2007 Birkhaueser.
English
DOI：https://doi.org/10.1007/s00030-007-5027-y
DOI ID：10.1007/s00030-007-5027-y, ISSN：1021-9722, SCOPUS ID：38849149769 - Dirac equation with certain quadratic nonlinearities in one space dimension
Shuji Machihara
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, Volume：9, Number：3, First page：421, Last page：435, Jun. 2007
We discuss the time local existence of solutions to the Dirac equation for special types of quadratic nonlinearities in one space dimension. Solutions with more rough data than those of the previous work [15] are obtained. The Fourier transforms of solutions with respect to both variables x and t are investigated. Certain linear and bilinear estimates on solutions are derived, and a standard iteration argument gives the existence results.
WORLD SCIENTIFIC PUBL CO PTE LTD, English
DOI：https://doi.org/10.1142/S0219199707002484
DOI ID：10.1142/S0219199707002484, ISSN：0219-1997, Web of Science ID：WOS:000251009000006 - The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation
Shuji MacHihara; Takayuki Omoso
Ricerche di Matematica, Volume：56, Number：1, First page：19, Last page：30, Jun. 2007
In [3] Dias and Figueira have reported that the square of the solution for the nonlinear Dirac equation satisfies the linear wave equation in one space dimension. So the aim of this paper is to proceed with their work and to clarify a structure of the nonlinear Dirac equation. The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation are obtained. © 2007 Springer-Verlag.
English
DOI：https://doi.org/10.1007/s11587-007-0002-9
DOI ID：10.1007/s11587-007-0002-9, ISSN：0035-5038, SCOPUS ID：34848894824 - Dirac equation with certain quadratic nonlinearities in one space dimension
Shuji Machihara
Communications in Contemporary Mathematics, Volume：9, Number：3, First page：421, Last page：435, Jun. 2007
We discuss the time local existence of solutions to the Dirac equation for special types of quadratic nonlinearities in one space dimension. Solutions with more rough data than those of the previous work [15] are obtained. The Fourier transforms of solutions with respect to both variables x and t are investigated. Certain linear and bilinear estimates on solutions are derived, and a standard iteration argument gives the existence results. © World Scientific Publishing Company.
English
DOI：https://doi.org/10.1142/S0219199707002484
DOI ID：10.1142/S0219199707002484, ISSN：0219-1997, SCOPUS ID：34347354116 - Bilinear estimates for the transport equations
Shuji Machihara
Volume：47, Number：1, First page：189, Last page：196, 2007 - The Cauchy problem for the 1-D Dirac-Klein-Gordon equation
Shuji Machihara
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, Volume：14, Number：5-6, First page：625, Last page：641, 2007
The Cauchy problem for the Dirac-Klein-Cordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein-Cordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with.
BIRKHAUSER VERLAG AG, English
DOI：https://doi.org/10.1007/s00030-007-5027-y
DOI ID：10.1007/s00030-007-5027-y, ISSN：1021-9722, Web of Science ID：WOS:000252402400009 - Bilinear estimates for the transport equations
Shuji Machihara
Advanced Studies in Pure Mathematics-Asymptotic Analysis and Singularities, Volume：47, Number：1, First page：189, Last page：196, 2007 - One dimensional Dirac equation with quadratic nonlinearities
S Machihara
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, Volume：13, Number：2, First page：277, Last page：290, Jul. 2005
The local well-posedness for the nonlinear Dirac equation with special forms of quadratic nonlinearities in one space dimension is obtained by two approaches. One is to apply the Fourier restriction norm method of Bourgain [2, 3] by showing the bilinear estimates for the nonlinearities. Another is to study the explicit solutions for wave equations and derive another bilinear estimates similar with Bournaveas [4].
AMER INST MATHEMATICAL SCIENCES, English
ISSN：1078-0947, Web of Science ID：WOS:000228933400003 - Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation
S Machihara; M Nakamura; K Nakanishi; T Ozawa
JOURNAL OF FUNCTIONAL ANALYSIS, Volume：219, Number：1, First page：1, Last page：20, Feb. 2005
We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposed-ness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable. (C) 2004 Elsevier Inc. All fights reserved.
ACADEMIC PRESS INC ELSEVIER SCIENCE, English
DOI：https://doi.org/10.1016/j.jfa.2004.07.005
DOI ID：10.1016/j.jfa.2004.07.005, ISSN：0022-1236, Web of Science ID：WOS:000226268200001 - One dimensional Dirac equation with quadratic nonlinearities
Shuji Machihara
Discrete and Continuous Dynamical Systems - Series A, Volume：13, First page：277, Last page：290, 2005 - SMALL GLOBAL SOLUTIONS FOR NONLINEAR DIRAC EQUATIONS
Shuji Machihara; Makoto Nakamura; Tohru Ozawa
DIFFERENTIAL AND INTEGRAL EQUATIONS, Volume：17, Number：5-6, First page：623, Last page：636, May 2004
The global Cauchy problem for nonlinear Dirac and Klein-Gordon equations in space-time Rn+1 is studied in Sobolev and Besov spaces. Global existence of small solutions is proved under a scale-invariant setting when reduced to the corresponding massless case.
KHAYYAM PUBL CO INC, English
ISSN：0893-4983, Web of Science ID：WOS:000208532700009 - The inviscid limit for the complex Ginzburg-Landau equation
S Machihara; Y Nakamura
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Volume：281, Number：2, First page：552, Last page：564, May 2003
We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrodinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations. (C) 2003 Elsevier Science (USA). All rights reserved.
ACADEMIC PRESS INC ELSEVIER SCIENCE, English
DOI：https://doi.org/10.1016/S0022-247X(03)00143-4
DOI ID：10.1016/S0022-247X(03)00143-4, ISSN：0022-247X, Web of Science ID：WOS:000183591300011 - The inviscid limit for the complex Ginzburg-Landau equation
S Machihara; Y Nakamura
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Volume：281, Number：2, First page：552, Last page：564, May 2003
We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrodinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations. (C) 2003 Elsevier Science (USA). All rights reserved.
ACADEMIC PRESS INC ELSEVIER SCIENCE, English
DOI：https://doi.org/10.1016/S0022-247X(03)00143-4
DOI ID：10.1016/S0022-247X(03)00143-4, ISSN：0022-247X, Web of Science ID：WOS:000183591300011 - Interpolation inequalities in Besov spaces
Shuji Machihara; Tohru Ozawa
Proceedings of the American Mathematical Society, Volume：131, Number：5, First page：1553, Last page：1556, May 2003
In this paper we present an interpolation inequality in the homogeneous Besov spaces on ℝn, which reduces to a number of well-known inequalities in special cases.
English
DOI：https://doi.org/10.1090/S0002-9939-02-06715-1
DOI ID：10.1090/S0002-9939-02-06715-1, ISSN：0002-9939, SCOPUS ID：0037408055 - Interpolation inequalities in Besov spaces
S Machihara; T Ozawa
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume：131, Number：5, First page：1553, Last page：1556, 2003
In this paper we present an interpolation inequality in the homogeneous Besov spaces on R-n, which reduces to a number of well-known inequalities in special cases.
AMER MATHEMATICAL SOC, English
DOI：https://doi.org/10.1090/S0002-9939-02-06715-1
DOI ID：10.1090/S0002-9939-02-06715-1, ISSN：0002-9939, Web of Science ID：WOS:000180467000026 - Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation
S Machihara; K Nakanishi; T Ozawa
REVISTA MATEMATICA IBEROAMERICANA, Volume：19, Number：1, First page：179, Last page：194, 2003
In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space H-s. We prove the existence and uniqueness of global solutions for small data in H-s with s > 1. The method of proof is based on the Strichartz estimate of L-t(2) type for Dirac and Klein-Gordon equations. We also prove that the solutions of the nonlinear Dirac equation after modulation of phase converge to the corresponding solutions of the nonlinear Schrodinger equation as the speed of light tends to infinity.
UNIVERSIDAD AUTONOMA MADRID, English
ISSN：0213-2230, Web of Science ID：WOS:000184289400007 - Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations
S Machihara; K Nakanishi; T Ozawa
MATHEMATISCHE ANNALEN, Volume：322, Number：3, First page：603, Last page：621, Mar. 2002
We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schrodinger equation in the energy space, after the infinite oscillation in time is removed. We also derive the optimal rate of convergence in L-2.
SPRINGER-VERLAG, English
DOI：https://doi.org/10.1007/s002080100293
DOI ID：10.1007/s002080100293, ISSN：0025-5831, Web of Science ID：WOS:000175046400009 - Small data global solutions for Dirac--Klein--Gordon equation
Shuji Machihara
Volume：15, First page：1511, Last page：1517, 2002 - Small data global solutions for Dirac--Klein--Gordon equation
Shuji Machihara
Differential and Integral Equations, Volume：15, First page：1511, Last page：1517, 2002 - The Nonrelativistic Limit of the Nonlinear Klein--Gordon Equation
Shuji Machihara
Volume：44, First page：243, Last page：252, 2001 - The Nonrelativistic Limit of the Nonlinear Klein--Gordon Equation
Shuji Machihara
Funkcialaj Ekvacioj, Volume：44, First page：243, Last page：252, 2001
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