Shuji Machihara

Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation

In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space Hs. We prove the existence and uniqueness of global solutions for small data in Hs with s > 1...

The inviscid limit for the complex Ginzburg-Landau equation

Abstract 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.

DIRAC EQUATION WITH CERTAIN QUADRATIC NONLINEARITIES IN ONE SPACE DIMENSION

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.

Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces

In this paper, we establish the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces. More precisely, we generalize the Hardy type inequality obtained in Edmunds and Triebel (Math. Nachr. 207:79-92, 1999). The generalized inequality allows us to take the exponents appearing in the inequality more flexibly, and its optimality is discussed in detail. O’Neil’s inequality and its reverse play an essential role for the proof.MSC:46E35, 26D10.

Scattering theory for the Dirac equation with a non-local term

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.

Global wellposedness for a one-dimensional Chern–Simons–Dirac system in Lp

ABSTRACT The global wellposedness in Lp(ℝ) for the Chern–Simons–Dirac equation in the 1+1 space and time dimension is discussed. We consider two types of quadratic nonlinearity: the null case and the non-null case. We show the time global wellposedness for the Chern–Simon–Dirac equation in the framework of Lp(ℝ), where 1≤p≤∞ for the null case. For the scaling critical case, p = 1, mass concentration phenomena of the solutions may occur in considering the time global solvability. We invoke the Delgado–Candy estimate which plays a crucial role in preventing concentration phenomena of the global solution. Our method is related to the original work of Candy (2011), who showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in the critical space L2(ℝ).

Notes on the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’

*Correspondence: wadade@se.kanazawa-u.ac.jp 3Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan Full list of author information is available at the end of the article The purpose of this note is to clarify the novelty of the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’ which was published in the J. Inequal. Appl. :,  []. After this paper was published, the authors were informed of the references [–], and [], the results of which partly overlap with those of []. In the paper [], the authors established the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces H n p p,q(R); see Section  in [] for the precise definition of H n p p,q(R). The main theorem in [] is stated as follows. Theorem A [, Theorem .] Let n ∈ N,  < p <∞,  < q≤∞ and  < α,β <∞. Then the inequality (∫