## Journals

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### Open Access Journals

DCDS-B

A damped stochastic beam equation driven by a Non-Gaussian Lévy
process is studied. Under appropriate conditions, the existence
theorem for a unique global weak solution is given. Moreover, we
also show the existence of a unique invariant measure associated
with the transition semigroup under mild conditions.

DCDS-B

In this paper, we consider the compressible magnetohydrodynamic equations with nonnegative thermal conductivity and electric conductivity. The coefficients of the viscosity, heat conductivity and magnetic diffusivity depend on density and temperature. Inspired by the framework of [11], [13] and [15], we use the maximal regularity and contraction mapping argument to prove the existence and uniqueness of local strong solutions with positive initial density in the bounded domain for any dimension.

DCDS

In this paper, we consider the initial-value problem for the
Degasperis-Procesi equation with a linear dispersion, which is an
approximation to the incompressible Euler equation for shallow water
waves. We establish local well-posedness and some global
existence of solutions for certain initial profiles and determine the
wave breaking phenomena for the equation. Finally,
we verify the occurrence of the breaking waves by numerical simulations.

DCDS

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.

DCDS-B

In this paper, we consider compressible Navier-Stokes-Korteweg
(N-S-K) equations with more general pressure laws, that is the pressure $P$ is non-monotone. We prove the stability of
weak solutions in the periodic domain $\Omega=\mathbb{T}^{N}$, when
$N = 2,3$. Utilizing an interesting Sobolev inequality to tackle the
complicated Korteweg term, we obtain the global existence of weak
solutions in one dimensional case. Moreover, when the initial data
is compactly supported in the whole space $\mathbb{R}$, we prove the compressible N-S-K equations will blow-up in finite time.

keywords:
global existence
,
stability
,
compressible Navier-Stokes equations
,
Korteweg type
,
blow-up.

DCDS-B

The ocean thermohaline circulation, also called meridional overturning
circulation, is caused by water density contrasts. This circulation has
large capacity of carrying heat around the globe and it thus affects the energy
budget and further affects the climate. We consider a thermohaline circulation
model in the meridional plane under external wind forcing. We show that,
when there is no wind forcing, the stream function and the density fluctuation
(under appropriate metrics) tend to zero exponentially fast as time goes to
infinity. With rapidly oscillating wind forcing, we obtain an averaging principle
for the thermohaline circulation model. This averaging principle provides
convergence results and comparison estimates between the original thermohaline
circulation and the averaged thermohaline circulation, where the wind
forcing is replaced by its time average. This establishes the validity for using
the averaged thermohaline circulation model for numerical simulations at long
time scales.

DCDS-B

Three dimensional primitive equations with a
small multiplicative noise are studied in this paper. The existence
and uniqueness of solutions with small initial value in a fixed
probability space are obtained. The proof is based on Galerkin
approximation, Itô's formula and weak convergence methods.

DCDS

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.

DCDS-B

This article is concerned with a mutualism ecological model with Lévy noise. The local existence and
uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior
to the problem is studied. Moreover, we show that the solution is stochastically bounded and stochastic permanence.
The sufficient conditions for the system to be extinct are given and the conditions for the system
to be persistence in mean are also established.

keywords:
Itô's formula
,
extinction
,
persistent in mean
,
stochastic permanence.
,
mutualism model

DCDS

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.

## Year of publication

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