Geometric invariant, propagation of singularity and asymptotic behavior for nonlinear wave equations

众所周知，非线性波动方程解的正则性与几何不变量之间有着密切的关系。特别是Klainman和Christodoulou提出的零条件在相对论非线性波动方程中往往起着重要的作用。从这个角度出发，我们在2000年和2001年研究了Dirac-Proca方程和Maxwell-Higgs方程柯西问题解的整体存在性。我们首先证明了Proca方程具有零条件结构，从而证明了对于较小且光滑的初值，解的整体存在性。其次，我们发现Maxwell-Higgs方程一般具有零条件结构，这使得我们能够证明小而光滑的初值的整体解的存在性。在2002学年，我们研究了弱类中修正的KdV方程柯西问题的适定性。我们知道，当S和lt；1/2时，解映射不在C^2中，而对于S和lt；1/2时，解映射在C^∞中。我们研究了修正的KdV方程的什么样的结构打破了弱类中的适定性

It is well known that there are close relations between the regularity of solutions and the geometric invariant for nonlinear wave equations. Especially, the null condition introduced by Klainerman and Christodoulou often plays an important role in the case of relativistic nonlinear wave equations. From this point of view, in the acadimic years of 2000 and 2001, we studied the global existence of solutions for the Cauchy problem of the Dirac-Proca equations and the Maxwell-Higgs equations. We first showed that the Proca equation has a null condition structure, which led to the global existence of solution for small and smooth initial data. We next discovered that the Maxwell-Higgs equations generically have a null condition structure, which enabled us to show the global existence of solution for small and smooth initial dataIn the academic year of 2002, we studied the well-posedness of the Cauchy problem for the modified KdV equation in a weak class. It is known that when s< 1/2, the solution map is not in C^2, while it is in C^∞ for s > 1/2. We studied what kind of structure for the modified KdV equation breaks down the well-posedness in a weak class