The Cauchy problem for the 2D magnetohydrodynamic-α equations

In this paper, we consider the Cauchy problem for the 2D incompressible magnetohydrodynamics-$\alpha$ (MHD-$\alpha$) equations. We obtain the global solution for the 2D incompressible MHD-$\alpha$ simulation model in the fractional index Sobolev space, and prove that the incompressible MHD-$\alpha$ equations reduce to the incompressible homogeneous MHD equations as $\alpha \to 0$, and the solution of MHD-$\alpha$ equations will converge to the weak solution of the corresponding MHD equations. Moreover, it is convenient to construct a numerical algorithm based on an iteration scheme in our proof.     

On pattern formation in the Gray-Scott model

In the paper, we investigate an elliptic system well-known as the Gray-Scott model and present some further results for positive solutions of this model. More precisely, we give the refined a priori estimates of positive solutions, and improve some previous results for the non-existence and existence of positive non-constant solutions as the parameters are varied, which imply some certain conditions where the pattern formation occurs or not.     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

On the approximation of the solutions of stochastic equations with θ-integrals

The paper investigates the problem of approximation of stochastic θ-integrals and the solutions of stochastic differential equations. The complete classification of the methods of approximation of stochastic θ-integrals in the convolution algebra is proposed. It is proved that the solutions of stochastic integral equations with θ-integral can be approximated by the solutions of finite-difference equations with averaging.     

The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process,which is a stochastic fluid model itself

We introduce the Stochastic Fluid–Fluid Model, which offers powerful modeling ability for a wide range of real-life systems of significance. We first derive the infinitesimal generator, with respect to time, of the driving stochastic fluid model. We then use this to derive the infinitesimal generator of a particular Laplace–Stieltjes transform of the model, which is the foundation of our analysis. We develop expressions for the Laplace–Stieltjes transforms of various performance measures for the transient and limiting analysis of the model. This work is the first direct analysis of a stochastic fluid model that is Markovian on a continuous state space.     

A stochastic modified Beverton–Holt model with the Allee effect

In this paper, we consider a discrete stochastic Beverton–Holt model with the Allee effect. We study the effects of demographic and environmental fluctuations on the dynamics of the model. Moreover, we investigate the potential function, the attainment time and quasi-stationary distributions of the system.     

Algebraic solution of the Stein–Stein model for stochastic volatility

We provide an algebraic approach to the solution of the Stein–Stein model for stochastic volatility which arises in the determination of the Radon–Nikodym density of the minimal entropy of the martingale measure. We extend our investigation to the case in which the parameters of the model are time-dependent. Our algorithmic approach obviates the need for Ansätze for the structure of the solution.     

Positivity-preserving symplectic methods for the stochastic Lotka–Volterra predator-prey model

BIT Numerical Mathematics - In this paper, positivity-preserving symplectic numerical approximations are investigated for the 2d-dimensional stochastic Lotka–Volterra predator-prey model...     

On the well-posedness of 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3], Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. $1<p<6$. The main novelty of this work is to apply the CRW theorem obtained by Coifman, Rochberg, Weiss in [11] to get a new a priori estimate for an elliptic equation with variable coefficients. The uniqueness of the solution also relies on a Lagrangian approach as in [16], [17], [18].     

On the high temperature region of the Sherrington–Kirkpatrick model

We prove the validity of the replica-symmetric formulas for the Sherrington–Kirkpatrick (SK) model in a region of parameters that (probably) coincides with the region predicted by the physicists.     

On the ground states of the one-dimensional Falicov–Kimball model

The ground states of the one-dimensional Falicov–Kimball model are studied in the grand canonical ensemble for large values of the interaction strength U. The quantum particle chemical potential μ_e is chosen in the interval −U+4<μ_e<0, such that, then, these states are neutral states and depend only on the sum of the two chemical potentials, μ=μ_i+μ_e. Consequences of this study are, among others, the following results. If ρ=p/q (p and q relatively prime) is a rational number we prove that, for U⩾U₀(q) (where U₀(q) is a specific function), there is an interval on the μ-axis, of length larger than qU^−2q+3, such that for any μ in this interval, the ground state has density ρ. In this interval the ground state is unique, up to translations, and the corresponding classical particle configuration is described by the characteristic sequence associated with the rational number ρ.     

On the Markov-modulated insurance risk model with tax

In this paper, we consider the Markov-modulated insurance risk model with tax. We assume that the claim inter-arrivals, claim sizes and premium process are influenced by an external Markovian environment process. The considered tax rule, which is the same as the one considered by Albrecher and Hipp [Blätter DGVFM 28(1):13–28, 2007], is to pay a certain proportion of the premium income, whenever the insurer is in a profitable situation. A system of differential equations of the non-ruin probabilities, given the initial environment state, are established in terms of the ruin probabilities under the Markov-modulated insurance risk model without tax. Furthermore, given the initial state, the differential equations satisfied by the expected accumulated discounted tax until ruin are also derived. We also give the analytical expressions for them by iteration methods.     

On symmetrized stochastic convexity and the inequalities of Hermite–Hadamard type

Aequationes mathematicae - In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a...     

On the global well-posedness of 3-D axi-symmetric Navier–Stokes system with small swirl component

In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier–Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the initial swirl component of the velocity being sufficiently small in the almost critical spaces, we can still prove the global well-posedness of the system.     

On unbiased stochastic Navier–Stokes equations

A random perturbation of a deterministic Navier?CStokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term ${u{\nabla}u}$ . This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier?CStokes equation. The perturbed equation is solved in the space of generalized stochastic processes using the Cameron?CMartin version of the Wiener chaos expansion. It is shown that the generalized solution is a Markov process and scales effectively by Catalan numbers.     

On the existence of the weak solution with local energy inequality to the 3-D inhomogeneous incompressible Navier–Stokes equations

In this paper, we show there exists a weak solution to the 3-D inhomogeneous incompressible Navier–Stokes equations satisfying in addition the local energy inequality. The same result in the homogeneous incompressible case was described in [1, Theorem 3.2].