论文标题
一类弱耗散的Camassa-Holm方程的全球溶解性和爆炸的新结果
New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations
论文作者
论文摘要
在本文中，我们考虑了一类无均匀贝贝空间中一类弱耗散的Camassa-Holm方程的凯奇问题。首先，我们证明凯奇问题在besov空间中接受了一个独特的全球强解决方案，其耗散参数$λ> 0 $。证明中的新成分在于将方程式转换为一类阻尼的Camassa-Holm方程，并执行非标准的迭代方法。结果表明，我们的结果适用于带有更一般时间依赖性参数的阻尼方程，这改善了存在的结果来自Sobolev空间可besov空间，而无需在初始数据上假设任何符号条件。其次，我们在合适的Sobolev空间中得出了两种爆炸标准，在某种意义上说，这会告诉我们耗散参数$λ$如何影响强解决方案的奇异性形成。
In this paper, we consider the Cauchy problem for a class of weakly dissipative Camassa-Holm equations in nonhomogeneous Besov spaces. First, we prove that the Cauchy problem admits a unique global strong solution in Besov spaces with proper condition on the dissipation parameter $λ>0$. The novel ingredients in the proof lies in transforming the equations into a class of damped Camassa-Holm equations, and performing a non-standard iterative method. It is shown that our result holds for the damped equations with more general time-dependent parameters, which improves the existed results from Sobolev spaces to Besov spaces without assuming any sign condition on the initial data. Second, we derive two kinds of blow-up criteria in suitable Sobolev spaces, which in some sense inform us how the dissipation parameter $λ$ influences the singularity formation of strong solutions.
