Research on propagation of singularities for nonlinear wave equations

1.首席研究员和共同研究员T.Ogawa与E.Kaikina和P. I. Naumkin一起研究了Benjamin-Ono方程的解在t>0的空间和时间变量中解析的充分条件。当初值在原点具有奇异性时，证明了当t> 0.2时解是解析的。本文研究了Benjamin-Ono方程初值问题解的存在性。然而Picard迭代法在通常的Sobolev空间中不可能应用于Benjamin-Ono方程，我们证明了Picard迭代法在某些具有齐次和非齐次乘子的Sobolev空间中是适用的。3.研究了非线性项满足零条件的非线性波动方程的奇性传播。在非线性项为零的条件下，正则性估计得到了改进。

1.The head investigator and co-investigator T.Ogawa has studied on the sufficient condition for solutions to the Benjamin-Ono equation to be analytic in space and time variables for t>0 with E.Kaikina and P.I.Naumkin. Under the condition that initial data have some singularity at the origin, we show that the solution is analytic for t>0.2.We have investigated the existence of solutions for the initial value problem of the Benjamin-Ono equation. However it is impossible to apply Picard's iteration method to the Benjamin-Ono equation in usual Sobolev space, we show that Picard's iteration method is applicable for some Sobolev space with homogeneous and inhomogeneous multipliers.3.We have investigate the propagation of singularities for the nonlinear wave equations with nonlinear term satisfying null condition. Regularity estimate can be improved under the null condition for nonlinear term.