Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.

Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform＊

This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail.

A BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE QUANTUM HYDRODYNAMIC MODEL

In this article,we focus on the short time strong solution to a compressible quantum hydrodynamic model.We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity,the second spacial derivative of the square root of the density,and the first order time derivative and first order spacial derivative of the square root of the density.

THEORETICAL RESULTS ON THE EXISTENCE,REGULARITY AND ASYMPTOTIC STABILITY OF ENHANCED PULLBACK ATTRACTORS:APPLICATIONS TO 3D PRIMITIVE EQUATIONS

Several new concepts of enhanced pullback attractors for nonautonomous dynamical systems are introduced here by uniformly enhancing the compactness and attraction of the usual pullback attractors over an infinite forward time-interval under strong and weak topologies.Then we provide some theoretical results for the existence,regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which can be applied to a large class of PDEs.The existence of these enhanced attractors is harder to obtain than the backward case[33],since it is difficult to uniformly control the long-time pullback behavior of the systems over the forward time-interval.As applications of our theoretical results,we consider the famous 3D primitive equations modelling the large-scale ocean and atmosphere dynamics,and prove the existence,regularity and asymptotic stability of the enhanced pullback attractors in V×V and H^(2)×H^(2) for the time-dependent forces which satisfy some weak conditions.

HIGH-ORDER NUMERICAL METHOD FOR SOLVING A SPACE DISTRIBUTED-ORDER TIME-FRACTIONAL DIFFUSION EQUATION

This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation.First,we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives.Second,based on the piecewise-quadratic polynomials,we construct the nodal basis functions,and then discretize the multi-term fractional equation by the finite volume method.For the time-fractional derivative,the finite difference method is used.Finally,the iterative scheme is proved to be unconditionally stable and convergent with the accuracy O(σ^(2)+τ^(2-β)+h^(3)),whereτand h are the time step size and the space step size,respectively.A numerical example is presented to verify the effectiveness of the proposed method.

INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL

This article studies the initial-boundary value problem for a three dimensional magnetic-curvature-driven Rayleigh-Taylor model.We first obtain the global existence of weak solutions for the full model equation by employing the Galerkin’s approximation method.Secondly,for a slightly simplified model,we show the existence and uniqueness of global strong solutions via the Banach’s fixed point theorem and vanishing viscosity method.

Global weak solutions to a phase-field model for motion of grain boundaries

We employ the Galerkin method to prove the global existence of weak solutions to a phase-field model which is suitable to describe a sort of interface motion driven by configurational forces.The higher-order derivative of unknown S exists in the sense of local weak derivatives since it may be not summable over the original open domain.The existence proof is valid in the one-dimensional case.

Global solution for quantum Zakharov equations

The initial value problem for the quantum Zakharov equation in three di- mensions is studied. The existence and uniqueness of a global smooth solution are proven with coupled a priori estimates and the Galerkin method.

THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM

This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries(KdV)system in Rn.We first prove,by the Leray-Schauder principle and the vanishing viscosity method,that any initial data N-dimensional vector value function u0(x)in Sobolev space H^(s)(R^(n))(s≥1)leads to a global weak solution.Second,we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in R^(2)and R^(3).

GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS

Global in time weak solutions to the α-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to α-model regularization for the three dimension compressible EulerPoisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies γ >4/3.

Unconditional convergence of a linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation

This paper is concerned with the time-step condition of a linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a 'lifting' technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schr¨odinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

Existence and blow-up of the solutions to the viscous quantum magnetohydrodynamic nematic liquid crystal model

In this paper, we investigate the coupled viscous quantum magnetohydrodynamic equations and nematic liquid crystal equations which describe the motion of the nematic liquid crystals under the magnetic field and the quantum effects in the two-dimensional case. We prove the existence of the global finite energy weak solutions by use of a singular pressure close to vacuum. Then we obtain the local-in-time existence of the smooth solution. In the final, the blow-up of the smooth solutions is studied. The main techniques are Faedo-Galerkin method, compactness theory, Arzela-Ascoli theorem and construction of the functional differential inequality.

Landau-Lifshitz-Bio ch equation on Riemannian manifold

We bring in Landau-Lifshitz-Bloch equation on m-dimensional closed Riemannian manifold and prove that it admits a unique local solution. When m ≥ 3 and the initial data in L^∞-norm is sufficiently small, the solution can be extended globally. Moreover, for m = 2, we can prove that the unique solution is global without assuming small initial data.

GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR FOR THE STRONG SOLUTIONS IN H2 TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper, we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in threedimensional whole space. The global existence of strong solutions is obtained by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H2-framework. If the initial datas in Ll-norm are finite additionally, the optimal time decay rates of strong solutions are established. With the help of Fourier splitting method, one also establishes optimal time decay rates for the higher order spatial derivatives of director.

OPTIMAL DECAY RATE OF THE COMPRESSIBLE QUANTUM NAVIER-STOKES EQUATIONS

For quantum fluids governed by the compressible quantum Navier-Stokes equations in R;with viscosity and heat conduction, we prove the optimal L;- L;decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.

Well-posedness and dynamics of fractional Fitz Hugh-Nagumo systems on R^(N) driven by nonlinear noise

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous,non-local,fractional,stochastic Fitz Hugh-Nagumo systems driven by nonlinear noise defined on the entire space RN.The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates.The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space.The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms.The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on RNas well as the lack of smoothing effect on one component of the solutions.The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

Local Well-posedness of the Derivative Schrödinger Equation in Higher Dimension for Any Large Data

In this paper,the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ut-iΔu+|u|^(2)(→γ·▽u)+u^(2)(→λ·▽-u)=0,(x,t)∈R^(n)×R,→γ,→λ∈R^(n);n≥2 It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in H^(s)(R^(n))(s>n/2)for any large initial data.Thus this result can compare with that in one dimension except for the endpoint space H^(n/2).

Shape Analysis of Bounded Traveling Wave Solutions and Solution to the Generalized Whitham-Broer-Kaup Equation with Dissipation Terms

This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short.The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ),H(ξ)),and then give its global phase portraits.The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions.The authors study the dissipation effect on the solutions,find out a critical value r*,and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater,i.e.,|r| ≥ r*,while it appears as a damped oscillatory wave if the dissipation effect is smaller,i.e.,|r| < r*.Two solitary wave solutions to the WBK equation without dissipation effect is also obtained.Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits,the authors obtain all approximate damped oscillatory solutions (u(ξ),H(ξ)) under various conditions by using the undetermined coefficients method.Finally,the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.

Instability of Standing Waves for Hamiltonian Wave Equations

This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest.On the one hand,by proving a compactness lemma and solving a variational problem,the existence of the standing wave with ground state for the aforementioned equation is proved.On the other hand,the authors derive the instability of the standing wave by applying the potential well argument,the concavity method and an invariant region under the solution flow of the Cauchy problem for the equation under study,and the invariance of the region aforementioned can be shown by introducing an auxiliary functional and a supplementary constrained variational problem.

The Cauchy problem for the stochastic generalized Benjamin-Ono equation

The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation.By establishing the bilinear and trilinear estimates in some Bourgain spaces,we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x,w)∈L^(2)(Ω;H^(s)(R))which is F0-measurable with s≥1/2-α/4 andΦ∈L20,s.In particular,whenα=1,we prove that it is globally well-posed for the initial data u0(x,w)∈L2(Ω;H1(R))which is F0-measurable andΦ∈L20,1.The key ingredients that we use in this paper are trilinear estimates,the Ito formula and the Burkholder-Davis-Gundy(BDG)inequality as well as the stopping time technique.