Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
Yong Chen Hongjun Gao María J. Garrido–Atienza Björn Schmalfuss
This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.
keywords: Stochastic PDEs fractional Brownian motion pathwise solutions random dynamical systems.
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
Yong Chen Hongjun Gao Yue Liu
Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
keywords: Two-component Dullin-Gottwald-Holm system global solutions regularization wave-breaking solitary-wave solutions.
Global existence for the stochastic Degasperis-Procesi equation
Yong Chen Hongjun Gao
This paper is concerned with the Cauchy problem of stochastic Degasperis-Procesi equation. Firstly, the local well-posedness for this system is established. Then the precise blow-up scenario for solutions to the system is derived. Finally, the gloabl well-posedness to the system is presented.
keywords: blow-up scenario Littlewood-Paley decomposition global well-posedness. Stochastic Degasperis-Procesi equation

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