Research on the Navier-Stokes equation and the related topics on the nonlinear differential equations

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Masuda considered the discrete Lax pair for discrete Toda equation. Flashka developed the inverse scattering method for the solution of the Toda equation. On the other hand, Hirota proposed time-discrete Toda equation. Hirota-Tsujimoto-Imai constructed the disrete Lax pair. But this Lax pair does not have symmery property. Masuda constructed symmetric discrete Lax pair in the frame work of funcitonal analysis.Masuda considered the degenerate nonlinear parabolic equation of the porous medium type and showed the comparison porperty for solutions of the degenerate nonlinear parabolic equations.Morimoto considered a steady Navier-Stokes equations on a 2-D bounded domain, symmetric with respect to the y-axis. The boundary has several connected components, intersecting the y-axis. The boundary value is symmetric with respect to the y-axis satisfying the general outflow condition. The existence of the symmetric solution to the steady Navier-Stokes equations was established by Amick and Fujita. Fujita proved a key lemma concerning the solenoidal extension of the boundary value by virtual method. Morimoto gave a proof by a method different from Fujita.Ishimura considered Eguchi-Oki-Matsumura equations which describes the dynamics of pattern formation that arises from phase separation in some binary alloys. The model extends the well-known Cahn-Hilliard equation and consists of coupled two functions ; one is the local concentration and the other is the local degree of order. Ishimura showed the existence of a solutions, its symptotic profile, and in part the structure of steady state solutions.Ishimura deals with the exact solutions for stagnation flows with slip/ The problem becomes the solvability of certain third-order differential equations(ODEs). Reducing the order of ODEs, Ishimura exhibit another elementary proof of the existence and asymptotic behavior of solutions.