论文标题
在三个空间维度中的某些准电波方程的非平凡全局解决方案
Nontrivial global solutions to some quasilinear wave equations in three space dimensions
论文作者
论文摘要
在本文中，我们试图在三个空间维度中为某些准线性波方程构建非平凡的全局解决方案。我们首先提出了针对准线性波方程的一般系统构建非平凡的全局解决方案的条件结果。假设存在对几何减少系统的全局解决方案存在并满足了几个精心挑选的估计值，我们找到了与原始波方程的匹配的精确全局解决方案。然后将这种条件结果应用于两种引起人们关注的方程式。一个是John's CounterExamples $ \ box u = u_t^2 $或$ \ box u = u_t u_ {tt} $，另一个是3D可压缩的Euler方程，没有涡流。我们明确地针对相应的几何减少系统构建全局解决方案，并表明这些全局解决方案满足了所需的点范围。结果，对于这两种方程式，存在着大量的非平凡全球解决方案。
In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of quasilinear wave equations. Assuming that a global solution to the geometric reduced system exists and satisfies several well-chosen pointwise estimates, we find a matching exact global solution to the original wave equations. Such a conditional result is then applied to two types of equations which are of great interest. One is John's counterexamples $\Box u=u_t^2$ or $\Box u=u_t u_{tt}$, and the other is the 3D compressible Euler equations with no vorticity. We explicitly construct global solutions to the corresponding geometric reduced systems and show that these global solutions satisfy the required pointwise bounds. As a result, there exists a large family of nontrivial global solutions to these two types of equations.
