与海啸相关的几个浅水波方程(组)的研究

In this project, we study some shallow water equations which may be relevant to tsunami- the Degasperis-Procesi(DP) equation, the b-family shallow water equation and the Dullin-Gottwald-Holm (DGH) system. Fistyly, we will continue our study on the wave -breaking phenomena and global strong solutions of the DGH system, as well as the behaviour of the strong solution after blowup occurs in finite time. Then we are going to study the orbital stability of the periodic peakon and multi-peakon solutions of the DP equation. Finally we turn our attention to the b-family eqution. We will study the solution map's H?lder continuity or even Lipschitz continuity in some Sobolev space in terms of the weak topology, after this, we will find what is a possible optimal H?lder's index

本项目主要关注与海啸相关的几个浅水波方程(组)-Degasperis-Procesi(DP)方程, b-族方程和Dullin-Gottwald-Holm方程组. 首先, 我们将对DGH方程组波的破裂现象和整体解进行更深入的研究, 之后将研究DGH方程组的强解在有限时刻爆破以后的行为.其次, 我们将研究DP方程周期尖峰解和多重尖峰解的轨道稳定性. 最后我们将研究 b-族方程的解对初值的H?lder甚至Lipschitz连续依赖性, 以及可能的最佳H?lder连续指标.

主要研究了水波方程领域中的几个有实际背景和物理意义的模型方程，如（generalized）Camassa-Holm (gCH)方程组、（generalized）modified Camassa-Holm (gmCH)方程、Degasperis-Procesi (DP)方程、（generalized）Dullin-Gottwald-Holm (gDGH)方程组、b族方程、Novikov方程等(强)解的适定性，如局部和整体存在性、有限时刻爆破的条件、爆破速度和稳定性等一系列问题。对DP方程、Novikov方程和generalized mCH方程建立了迄今为止最优的“局部”爆破条件；得到了gDGH方程组行波解的分类；给出了CH方程组有限时刻爆破的新条件；研究了b族方程的解在弱拓扑下的连续依赖性；证明了Novikov方程的peakon解在能量空间中的轨道稳定性。