组合KdV方程解在图灵可计算意义下的有界性

The Boundness of the Solutions of the Combined 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,研究组合KdV方程初值问题解的有界性.首先在Schwartz空间得到了方程解及解的任意阶导的上确界可以由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),所以组合KdV方程解在Hs(R)(s≥0)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组合KdV方程的解奠定了基础.

By using Gronwall inequality, mathematical induction and the conservation quantity,the boundedness of the solutions which satisfy the initial value problem of Combined KdV equation is studied. Least upper bound of the solutions is actually controlled by the Turing computable function whose initial value is a variable on the Schwartz space S（R）. Because the S（R） is dense in H^s（R）（s ≥ 0）, the results can be extended to H^s （R） straightly. So least upper bound of the solutions is controlled by the computable function on H^s（R）. Accordingly,it could lay the foundation for studying the computability of the solution operator and using the Turing machine to compute the solutions of combined KdV equation.