Theory of global well-posedness on the nonlinear partial differential equations

研究了任意维区域上强解的局部存在性及其在有限时间内的爆破。局部解的生存期用给定初始数据的L^1和L^p-范数来刻画。同时，阐明了初始数据的总质量和第二动量以及方程组的系数对爆破现象有很大的影响。作为应用，我们证明了爆破解要么具有由p决定的爆破速率，要么在L^1中以大于绝对常数的振幅振荡。此外，在多连通区域中，对于具有非齐次边界数据且总通量为零的定常Navier-Stoeks方程，是否存在解仍然是一个悬而未决的问题。方程的非线性结构与区域的拓扑不变性之间的关系对该问题的可解性起着重要的作用。我们证明了，如果与域的第二Betti数相关联的给定边界数据的螺线管延伸的调和部分正交于Euler方程的非平凡解，则存在任何粘性常数的解。还阐明了Leary不等式与整环拓扑类型之间的关系。

We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified.