一类几何流方程周期解的爆破

Blowup of periodic solutions for nonlinear equations of geometric flow

研究双曲平均曲率流中一类几何流方程周期解的爆破问题.引入合适的黎曼不变量,将该方程化为对角型的一阶拟线性双曲型方程组.该方程组在Lax意义下不是真正非线性的.假设初值是周期的,且在一个周期内全变差很小,此外假设初值还满足一定的结构条件,可以证得该几何流方程的周期解必在有限时间内发生爆破,解的生命跨度估计可以给出.

This article considers the blow up problem for a class of nonlinear partial differential equations of geometric flow. By introducing the proper Riemann invariants, the equations can be reduced into a system of quasilinear hyperbolic equations in the diagonal form, which are not genuinely nonlinear in the sense of Lax. Under the assumptions that the initial data have small total variations in one period and some certain conditions are satisfied, the C1 solutions can be proved to blow up in finite time. In addition, the life span of the C1 classical solutions are derived.