Study on Navier-Stokes equations and the related nonlinear differential equations

在2006-2007年期间，增田研究了描述合金中二元相变的模型在溶液中的动力学行为。该模型由Eguchi-Oki-Matsumura（1984）提出的一个四阶偏微分方程组描述，增田成功地证明了所谓的Eguch-Oki-Matsumra方程的最大吸引子和惯性集的存在性，并在雅典国际会议上报告了这一结果。该模型产生一个非线性抛物型偏微分方程，其非线性反映了交易费用的存在。Ishimura用解析方法证明了解的存在性，并给出了解的数值计算方法。数值计算表明了Ishimura提出的方案的有效性.Ishimura关心的是由随机风险过程下的最优投资问题导出的某些偏微分方程的可解性.该方程描述了绝对风险厌恶的Arrow-Pratt系数关于最优值函数的演化，利用不动点方法结合收敛性证明了解的存在性.

In the period 2006-2007, Masuda studied the dynamical behavior in time of solution of a model describing the phase transition in binary in alloys. This model is described by a system of partial differential equations of the forth order, proposed by Eguchi-Oki-Matsumura(1984).Masuda succeeded in showing the existence of maximal attractor and inertial set for the so-called Eguch-Oki-Matsumra Equation and reported the results in the International Congress held at Athens.The Hoggard-Whalley-Willmott equation is introduced to mode portforios of European type options incorporating transaction costs. The model gives rise to a nonlinear parabolic partial differential equation, whose nonlinearity reflects the presence of transaction costs. Ishimura showed analytically the existence of solutions which are deviced to effectively handle an infinite domain and unbounded solution.Also Ishimura deal with numerical computation of solution. Numerical computation shows the validity of the scheme proposed by Ishimura.Ishimura is concerned with the solvability of certain partial differential equations, which is derived from the optimal Investment problem under the random risk process. The equations describe the evolution of the Arrow-Pratt coefficient of absolute risk aversion woth respect to the optimal value function.Employing the fixed point approach combined with the convergence argument Ishimura shows the existence of solution.