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

Masuda考虑离散Toda方程的离散Lax对。Flashka提出了逆散射法求解Toda方程。另一方面，Hirota提出了时间离散的Toda方程。Hirota-Tsujimoto-Imai构造了离散的Lax对。但是这个Lax对不具有对称性。Masuda在泛函分析的框架下构造了对称离散Lax对。Masuda考虑了多孔介质类型的退化非线性抛物方程，并给出了退化非线性抛物方程解的比较性质。Morimoto考虑了二维有界域上的稳定Navier-Stokes方程，该方程相对于y轴对称。边界有几个相互连接的组件，与y轴相交。边界值相对于满足一般流出条件的y轴是对称的。Amick和Fujita建立了稳态Navier-Stokes方程对称解的存在性。Fujita用虚法证明了关于边值的螺线形扩展的一个关键引理。森本用一种不同于藤田的方法给出了证明。Ishimura考虑了Eguchi-Oki-Matsumura方程，该方程描述了某些二元合金中由相分离引起的图案形成动力学。该模型扩展了著名的Cahn-Hilliard方程，由两个耦合函数组成；一个是局部浓度，另一个是局部有序度。Ishimura证明了一个解的存在性，它的症状分布，以及部分稳态解的结构。Ishimura处理带滑移的停滞流动的精确解，问题变成了某些三阶微分方程的可解性。通过降低ode的阶数，Ishimura给出了解的存在性和渐近性的另一个初等证明。

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.