Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions
Hongjun Gao Chengfeng Sun
Discrete & Continuous Dynamical Systems - B 2016, 21(9): 3053-3073 doi: 10.3934/dcdsb.2016087
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.
keywords: Itô's formula uniqueness. Galerkin approximations existence Primitive equation
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
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 79-98 doi: 10.3934/dcds.2014.34.79
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 stochastic beam equation driven by a Non-Gaussian Lévy process
Hongjun Gao Fei Liang
Discrete & Continuous Dynamical Systems - B 2014, 19(4): 1027-1045 doi: 10.3934/dcdsb.2014.19.1027
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.
keywords: invariant measure. global weak solution Lévy process Stochastic beam equation transition semigroup
Local strong solutions to the compressible viscous magnetohydrodynamic equations
Tong Tang Hongjun Gao
Discrete & Continuous Dynamical Systems - B 2016, 21(5): 1617-1633 doi: 10.3934/dcdsb.2016014
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.
keywords: compressible magnetohydrodynamic. strong solutions Existence
On the initial-value problem to the Degasperis-Procesi equation with linear dispersion
Fei Guo Bao-Feng Feng Hongjun Gao Yue Liu
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1269-1290 doi: 10.3934/dcds.2010.26.1269
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.
keywords: Degasperis-Procesi equation Global existence. Blow-up Local well-posedness Breaking waves phenomena
Global existence for the stochastic Degasperis-Procesi equation
Yong Chen Hongjun Gao
Discrete & Continuous Dynamical Systems - A 2015, 35(11): 5171-5184 doi: 10.3934/dcds.2015.35.5171
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
On the compressible Navier-Stokes-Korteweg equations
Tong Tang Hongjun Gao
Discrete & Continuous Dynamical Systems - B 2016, 21(8): 2745-2766 doi: 10.3934/dcdsb.2016071
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.
Dynamics of the thermohaline circulation under wind forcing
Hongjun Gao Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2002, 2(2): 205-219 doi: 10.3934/dcdsb.2002.2.205
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.
keywords: wind forcing. exponential decay Geophysical flows averaging principle
Dynamics of a delay Schistosomiasis model in snail infections
Chunhua Shan Hongjun Gao Huaiping Zhu
Mathematical Biosciences & Engineering 2011, 8(4): 1099-1115 doi: 10.3934/mbe.2011.8.1099
In this paper we modify and study a system of delay differential equations model proposed by Nåsell and Hirsch (1973) for the transmission dynamics of schistosomiasis. The modified stochastic version of MacDonald’s model takes into account the time delay for the transmission of infection. We carry out bifurcation studies of the model. The saddle-node bifurcation of the model suggests that the transmission and spread of schistosomiasis is initial size dependent. The existence of a Hopf bifurcation due to the delay indicates that the transmission can be periodic.
keywords: time delay stability Hopf bifurcation. saddle-node bifurcation Schistosomiasis
Analysis of a non-autonomous mutualism model driven by Levy jumps
Mei Li Hongjun Gao Bingjun Wang
Discrete & Continuous Dynamical Systems - B 2016, 21(4): 1189-1202 doi: 10.3934/dcdsb.2016.21.1189
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

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