某些色散波方程的适定性问题

【作者】 霍朝辉；

【导师】 郭柏灵；

【作者基本信息】 中国工程物理研究院 ， 应用数学， 2004， 博士

【摘要】 本论文主要研究某些色散波方程Cauchy问题的适定性。所用方法是Fourier限制范数方法，此方法是由J．Bourgain引进的。需要强调的是，所讨论的方程的生成半群的相函数及其一阶，二阶导数有非零奇异点，这就带来一些新的困难。但是，可以利用Fourier限制算子来分离这些奇异点，这样就可以改进已知的一些结果。本论文主要分四章。 在第二章，主要讨论Korteweg-de Vries-Benjamin-Ono方程的适定性问题。证明了当初值属于H~s(R)(s≥-1/8)，其Cauchy问题是局部适定的。当初值属于L~2(R)，其Cauchy问题是整体适定的。 在第三章，主要讨论Hirota方程的Cauchy问题。当方程带有具有导数的非线性项时，证明了若初值属于H~s(R)(s≥1/4)，其Cauchy问题是局部适定的；若初值属于H~s(R)(s≥1)，其Cauchy问题是整体适定的。当方程不带有具有导数的非线性项时，证明了若初值属于H~s(R)(s≥-1/4)，其Cauchy问题是局部适定的；若初值属于H~s(R)(s≥0)，其Cauchy问题是整体适定性的。关于整体适定性的证明，主要思想是解在空间H~s(s≥1)(H~s(s≥0))中的存在区间只依赖于初值的H~1(L~2)模。 在第四章，主要讨论关于涡旋丝的四阶非线性Schrdinger方程的Cauchy问题。证明了当方程具有某些系数限制时，若初值属于H~s(R)(s≥1/2)，其Cauchy问题是局部适定的。更多还原

【Abstract】 In the dissertation, the Cauchy problems of some dispersive equations are considered by the Fourier restriction norm method, which was first introduced by J.Bourgain. It is important to point out that the phase functions, their first-order derivatives and second-order derivatives have non-zero singular points, which makes the problem much more difficult. However, we can overcome the difficulties by the Fourier restriction operators to separate these singular points. Therefore, some known results can be improved. The dissertation consists of four chapters.In Chapter 2, the Cauchy Problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local well-posedness for data in Hs(R)(s > -1/8) and global well-posedness for data in L2(R) are obtained.In Chapter 3, the Cauchy problem of Hirota equation is studied. For the equation with derivative in nonlinear terms, the Cauchy problem is locally well-posed for data in Hs(R){s > 1/4) and globally well-posed for data in Hs(R)(s > 1). For the equation without derivative, the Cauchy problem is locally well-posed for data in Hs(R)(s > -1/4) and globally well-posed for data in Hs(R)(s > 0). The main idea for the global well-posedness, based on the generalized trilinear estimates, is that the existence time of the solution in Hs(s > 1)( Hs(R)(s > 0)) only depends on the norm of initial data in H1 (L2).In Chapter 4, the Cauchy problem for the Fourth-order nonlinear Schrodinger equation related to the vortex filament space is considered. Local result for data in Hs(R)(s > 1/2) is obtained under certain coefficient condition.更多还原