一类特殊空间上映射的梯度几何流柯西问题解的存在唯一性

通过几何分析方法与抛物型方程组解的逼近理论,研究特殊空间(一维球面S~1到二维球面S~2)上映射的梯度几何流柯西问题解的存在唯一性.利用能量法和空间本身特有的性质来解决能量守恒的问题,并利用适当的抛物型方程组逼近该梯度几何流,在适当的Sobolev空间中建立先验估计,找到其时间的一致正下界和抛物型方程组一列解的Sobo1ev范数的一致边界,借助于抛物型偏微分方程的理论,以此决定该柯西问题解的存在唯一性.

Existence and uniqueness of solution to Cauchy problem for gradient geometric flow of maps into special space

By using the method of the geometrical analysis and the approximation theory of the parabolic equations,the existence and the uniqueness of the solution to the Cauchy problem for gradient geometric flow of maps into a special space (one-dimensional sphere S~1 into two-dimensional sphere S~2) are studied.Use the energy method and the special nature of space itself to solve the problem of energy conservation,use the appropriate parabolic equations to approach the geometric flow,and establish a priori estimate in a suitable Sobolev space to find a uniform positive lower bound of the time and uniform bounds of various Sobolev norms, with the help of the perfect theory of parabolic partial differential equations.The existence and the uniqueness of the solution to the Cauchy problem are presented.