Relations between geometric invariant and singularity of solution in nonlinear evolution equations related to nonlinear waves

在1997年7月至2000年3月期间，我们研究了以下两个主题。我们首先研究了具有不同传播速度的非线性波动方程组柯西问题的适定性。寻找具有二次非线性的波动方程适定的最大可能函数空间是偏微分方程组研究领域的一个重要问题。这个问题与波动方程的洛伦兹不变量密切相关。当我们考虑具有不同传播速度的非线性波动方程组时，传播速度的差异打破了洛伦兹对称。我们从时间局部适定性的角度对二次非线性项进行了分类。其次，我们研究了具有随机强迫项的Korteweg-de Vries方程柯西问题的唯一可解性。从数学的角度来看，随机强迫项被视为一种非光滑摄动。特别是，逆散射法一般不适用于含强迫项的KDV方程。我们研究了一类自然随机力的解的时间局部存在性。

We had studied the following two subjects for the period of July, 1997-March, 2000.We first studied the well-posedness of the Cauchy problem for the system of nonlinear wave equations with different propagation speeds. One of the most important problems in the field of partial differential equations is to look for the largest possible function space in which the wave equations with quadratic nonlinearity is well-posed. This problem is closely related to the Lorentz invariant for the wave equation. When we consider the system of nonlinear wave equations with different propagation speeds, the discrepancy of propagation speeds breaks the Lorentz symmetry. We classified the quadratic nonlinear terms from a point of view of the time local well-posedness.Second, we studied the unique solvability of the Cauchy problem for the Korteweg-de Vries equation with stochastic forcing term. The stochastic forcing term is regarded as a nonsmooth perturbation from a mathematical point of view. Especially, in general, the inverse scattering method is inapplicable to the Korteweg-de Vries equation with forcing term. We investigated the time local existence of solution for a natural class of stochastic forces.