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The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces H s with s > 3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.

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By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

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This paper is concerned with the periodic boundary problem for the b-family equation. At first, we use a different method to prove the local well-posedness in the critical Besov space B 3/2 2,1 . Then we show that if a weaker B q p,r -topology is used, the solution map becomes Hölder continuous. Moreover, we show that the dependence on initial data is optimal in B 3/2 2,1 in the sense that the solution map is continuous but not uniformly continuous. Finally, we obtain the periodic peaked solutions to the b-family equation and apply them to obtain the ill-posedness in B 3/2 2,∞ .

In this paper, we consider the Cauchy problem for a two-component Novikov equation in the critical Besov space B 5/2 2,1 . We first derive a new a priori estimate for the 1-D transport equation in B 3/2 2,∞ , which is the endpoint case. Then we apply this a priori estimate and the Osgood lemma to prove the local existence. Moreover, we also show that the solution map u 0 → u is Hölder continuous in B 5/2 2,1 equipped with weaker topology. It is worth mentioning that our method is different from the previous one that involves extracting a convergent subsequence from an iterative sequence in critical Besov spaces.

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