论文标题
在随机的Dullin-Gottwald-Holm方程中：全球存在和波浪破坏现象
On the stochastic Dullin-Gottwald-Holm equation: Global existence and wave-breaking phenomena
论文作者
论文摘要
我们考虑了一类随机演化方程，其中包括随机凸轮 - 霍尔姆方程。对于圆环上的初始值问题，我们首先在sobolev spaces $ h^s $中建立局部存在和唯一性，并使用$ s> 3/2 $。然后我们表明，足够强的非线性噪声几乎可以防止爆炸。为了分析噪声较弱的效果，我们考虑使用非自主因素的线性乘法噪声。然后，我们在固有数据上制定精确的条件，从而导致全球存在强大的解决方案或爆炸。随着波浪的破裂，爆炸发生。对于带有正概率的爆破，我们得出了这些概率的下限。最后，精确分析了这些解决方案的爆炸率。
We consider a class of stochastic evolution equations that include in particular the stochastic Camassa--Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$. Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the inital data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.
