Modern Analysis for the Equations of Mathematical Science

主要研究数学科学中微分方程组的解的结构和解空间的结构。(1)研究双曲型偏微分方程适定性的特征，特别是具有三重特征根的方程的有效双曲性。奇点沿着双特征线的传播（2）研究随机微分方程的解，特别是Donsker δ函数的Wiener混沌展开，利用Levi-sum方法从Gauss核构造热核。用Donsker δ函数逼近热核的方法（3）可解格点模型状态空间结构的研究。用量子群表示法给出了六顶点模型关联函数的显式表达式。（4）动力学系统中分岔与混沌的研究。具有三个零特征值的退化向量场的小扰动引起的几何奇异吸引子的出现。同宿或异宿轨道的余维二分支，特别是无穷多个同宿二重bif。(5)流体动力学方程的分歧问题。自由面问题定常分岔和Hopf分岔的计算机辅助证明

Main theme is the investigation of the structure of the solution and the structure of the solution space for the system of the differential equations in the mathematical science.(1) Investigation for the characterization of the wellposedness of the hyperbolic partial differential equation, specially the effective hyperbolicity for the equation which has triple chracteristic roots. Propagation of singularities along the bicharacteristics(2) Investigations of the solutions of stochastic differential equations, especially the expansion of the Donsker's delta function by the wiener chaos using the Levi-sum method which constructs the heat kernel from the Gauss kernel. Approximation scheme for the heat kernel using the Donsker's delta function(3) Investigation for the structure of the space of states for solvable lattice models. Explicit formula for the correlation function for the six-vertex model using the representation of quantum groups(4) Investigation of bifurcations and chaos in the Dynamical systems. Appearance of the geometric strange attractors by small perturbation of the degenerate vector field which has triple zero characteristic values. Codimension two bifurcations of homoclinic or heteroclinic orbits, specially infinitely many numbers of homoclinic doubling bif.(5) Bifurcation problems for the equations of fluid dynamics. Investigations and proof of the stationary and Hopf bifurcations for the free surface problems using the computer assisted proof