广义KdV方程解算子在图灵可计算意义下的有界性分析

On the Boundness of the Solution of Generalized KdV Equation under the Turing Computable Circumstance

运用数学归纳法、Gronwall不等式及方程的守恒量等工具研究并证明了广义KdV方程初值问题解的有界性.在Schwartz空间上得到了广义KdV方程的解,该方程解的任意阶导的上确界具有可控性,可通过初值为变量的图灵可计算函数来控制.由于Schwartz空间S(R)是Sobolev空间Hs(R)(s≥0)的稠子空间,结果可以直接推广到Sobolev空间Hs(R)(s≥0),所以广义KdV方程解在Hs(R)(s≥0)的上确界可以由一个可计算函数来控制,从而为研究解算子的可计算性并运用图灵机计算广义KdV方程的解奠定了基础.

The boundness of the solutions which satisfies the initiaI value problem of generalized KdV equation is studied. By using Gronwall inequality, mathematical induction and the conscrva tion quantity, least upper bound of the solutions is actually eontrolled by the Turing computable function whose initial value is a variable on the Schwartz space S（R）. As the S（R） is dense in H（ R ）（s≥0）, the results can be extended to H＇（ R ）（s≥0） straightly. So least upper bound of the solutions is controlled by the computable function on M＇（R）（s≥0）. Accordingly, it could lay the foundation for studying the computability of the solution operator and using the Turning machine to compute the solutions of generalized KdV equation.