Structure of Solutions and Geometric Symmetry for Nonlinear Evolution Equations

下面将介绍本课题的主要研究成果。在2003-4学年，我们研究了具有周期边界条件的修正KdV方程的Cauchy问题的时间局部适定性。1993年，Bourgain证明了这个柯西问题在$H^s$， s≧1/2中是时间局部适定的。此外，Bourgain（1996）也证明了解映射在$H^s$, s<1/2中不属于$C^3$，尽管在$H^s$， s≧1/2中是解析的。Takaoka和Tsutsumi对$H^s$， s≧1/2和$H^s$, s<1/2的区别进行了深入的研究。证明了柯西问题在$H^s$, s>1/3区间仍然是局部适定的，但由于非线性振荡的出现，解映射不是一致连续的。2005学年，Tsutsumi与Akihiro Shimomura一起研究了二维二次非线性薛定谔方程解的渐近行为。在二维空间中，二次非线性是介于近程和长距离之间的一个边界，从非线性散射理论的角度来看，二次非线性具有特殊的意义。结果表明，对于一个平方模量的未知非线性，其解不接近自由解，而已知的其他二次非线性则属于短程相互作用。混浊。2006学年，Tsutsumi研究了非线性薛定谔方程Cauchy问题解的无条件唯一性。当构造解时，通常会附加一个条件，即该解属于与Stricahrtz估计相关联的辅助空间。无条件唯一性是指解即使不在这种辅助空间中也是唯一的。证明了非线性薛定谔方程在临界Sobolev空间解的尺度不变性的无条件唯一性。这比furoli和Terraneo的结果有了很大的改进。少

In what follows, the main research results of this project are described.In the academic years of 2003-4, we studied the time local well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. In 1993, Bourgain proved that this Cauchy problem is time locally well-posed in $H^s$, s ≧1/2. Furthermore, in 1996, Bourgain also proved that the solution map fails to belong to $C^3$ in $H^s$, s<1/2, though it is analytic in $H^s$, s≧ 1/2. Takaoka and Tsutsumi made a close investigation into the problem of what is the difference between the cases $H^s$, s ≧1/2 and $H^s$, s<1/2. It was showed that the Cauchy problem is still locally well-posed in $H^s$, s>1/3, but the solution map is not uniformly continuous because of the occurrence of nonlinear oscillation.In the academic year of 2005, Tsutsumi studied the asymptotic behavior of solution for the quadratic nonlinear Schrodinger equation in two space dimensions with Akihiro Shimomura. In two space dimensio … More ns, the quadratic nonlinearity is a boader between the short renge case and the long range case and the quadratic nonlinearity has a special interest from a viewpoint of nonlinear scattering theory. It was showed that for a nonlinearity of squared modulous of the unkown, the solution does not approach a free solution, while it is already known that the rest of other quadratic nonlinearities belong to the shourt range interaction. casse.In the academic year of 2006, Tsutsumi studied the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation. When the solution is constructed, one usually impose a condition that the solution belongs to an auxiliary space associated with the Stricahrtz estimate. The unconditional uniqueness means that the solution is unique even though it is not in this kind of auxiliary space. It was proved that the unconditional uniqueness holds for the solution in the critical Sobolev space associated with the scaling invariance of nonlinear SchrOdinger equations. This is a substantial improvement over the results by Furioli and Terraneo. Less