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

在2006-2007年间，Masuda研究了一个描述合金中二元相变的模型在溶解时的动力学行为。该模型由Eguchi-Oki-MatSumura(1984)提出的四阶偏微分方程组来描述。Masuda成功地证明了所谓的EguchOki-MatSumra方程存在极大吸引子和惯性集，并在雅典举行的国际会议上报道了这一结果。Hoggard-Whalley-Willmott方程被引入到考虑交易费用的欧式期权模型中。该模型得到了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。石村用解析的方法证明了解的存在，从而有效地处理了无限域和无界解。同时，石村还研究了解的数值计算。数值计算表明了Ishimura提出的方法的有效性。Ishimura研究了随机风险过程下最优投资问题中的某些偏微分方程组的可解性问题。该方程描述了绝对风险厌恶的Arrow-Pratt系数相对于最优值函数的演化。利用不动点方法结合收敛论证，Ishimura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.