Stability of half-degree point defect profiles for 2-D nematic liquid crystal
Zhiyuan Geng Wei Wang Pingwen Zhang Zhifei Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6227-6242 doi: 10.3934/dcds.2017269

In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.

keywords: Liquid crystal point defect Landau-de Gennes model monotonicity stability
Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system
Ling-Hao Zhang Wei Wang
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 591-604 doi: 10.3934/dcds.2017024

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

keywords: Bifurcation hyperbolic function energy balance method strongly nonlinear
Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain
Wei Wang Anthony Roberts
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 253-273 doi: 10.3934/dcds.2011.31.253
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.
keywords: averaging overlapping finite elements interelement coupling. Stochastic reaction-diffusion equation
Limit behavior of nonlinear stochastic wave equations with singular perturbation
Wei Wang Yan Lv
Discrete & Continuous Dynamical Systems - B 2010, 13(1): 175-193 doi: 10.3934/dcdsb.2010.13.175
Dynamical behavior of the following nonlinear stochastic damped wave equations

$ \nu $utt$+u_t=$Δ$u+f(u)+$ε$\dot{W}$

on an open bounded domain $D\subset\R^n$, $1\leq n\leq 3$,, is studied in the sense of distribution for small $\nu, $ε$>0$. Here $\nu$ is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary solution is constructed which describes the asymptotic behavior of solution from initial value in state space $H_0^1(D)\times L^2(D)$. Then a global measure attractor is constructed for (1). Furthermore under the case that the stochastic force is proportional to the square root of singular perturbation, that is ε$=\sqrt{\nu}$, we study the limit of the behavior of all the stationary solutions of (1) as $\nu\rightarrow 0$. It is shown that, by studying a continuity property on $\nu$ for the measure attractors of (1), any one stationary solution of the limit equation


is a limit point of a stationary solution of (1), as $\nu\rightarrow 0$.

keywords: stationary measure Tightness measure attractor. stochastic partial differential equations
Closed trajectories on symmetric convex Hamiltonian energy surfaces
Wei Wang
Discrete & Continuous Dynamical Systems - A 2012, 32(2): 679-701 doi: 10.3934/dcds.2012.32.679
In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin, where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sigma$.
keywords: Hamiltonian systems. closed characteristics Compact convex hypersurfaces
On fuzzy filters of Heyting-algebras
Wei Wang Xiao-Long Xin
Discrete & Continuous Dynamical Systems - S 2011, 4(6): 1611-1619 doi: 10.3934/dcdss.2011.4.1611
The concept of fuzzy filter of Heyting-algebras was introduced and some important properties were discussed. Some special kinds of fuzzy filters were defined and we prove that fuzzy Boolean filter is equivelent to fuzzy implicative filter in Heyting-algebras. And the relation among the fuzzy filters were proposed.
keywords: Heyting-algebras positive implicative implicative fuzzy filter ultra fuzzy (prime obstinate) filters. Boolean
Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections
Wei Wang Wanbiao Ma
Discrete & Continuous Dynamical Systems - B 2018, 23(8): 3213-3235 doi: 10.3934/dcdsb.2018242

We consider a class of non-cooperative reaction-diffusion system, which includes different types of incidence rates for virus dynamical models with nonlocal infections. Threshold dynamics are expressed by basic reproduction number $ R_0 $ in the following sense, if $ R_0 < 1 $, the infection-free steady state is globally attractive, implying infection becomes extinct; while if $ R_0 > 1 $, virus will persist. To study the invasion speed of virus, the existence of travelling wave solutions is studied by employing Schauder's fixed point theorem. The method of constructing super-solutions and sub-solutions is very technical. The mathematical difficulty is the problem constructing a bounded cone to apply the Schauder's fixed point theorem. As compared to previous mathematical studies for diffusive virus dynamical models, the novelty here is that we successfully establish the general existence result of travelling wave solutions for a class of virus dynamical models with complex nonlinear transmissions and nonlocal infections.

keywords: Reaction-diffusion system nonlinear transmissions threshold dynamics travelling wave solutions nonlocal infections
On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals
Yuning Liu Wei Wang
Discrete & Continuous Dynamical Systems - B 2018, 23(9): 3879-3899 doi: 10.3934/dcdsb.2018115

This work is concerned with the solvability of a Navier-Stokes/Q-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

keywords: Liquid crystals Q-tensor Landau-de Gennes theory initial boundary problem local well-posedness
Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production
Mengyao Ding Wei Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-20 doi: 10.3934/dcdsb.2018328

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: $ u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v) $, $ \tau_1v_t = \Delta v-a_1v+b_1w $, $ \tau_2w_t = \Delta w-a_2w+b_2u $, under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\Bbb{R}^{n} $ ($ n\geq 1 $), where $ \tau_i,a_i,b_i>0 $ ($ i = 1,2 $) are constants, and the diffusivity $ D $ and the density-dependent sensitivity $ S $ satisfy $ D(s)\geq a_0(s+1)^{-\alpha} $ and $ 0\leq S(s)\leq b_0(s+1)^{\beta} $ for all $ s\geq 0 $ with $ a_0,b_0>0 $ and $ \alpha,\beta\in\Bbb R $. It is proved that if $ \alpha+\beta<3 $ and $ n = 1 $, or $ \alpha+\beta<4/n $ with $ n\geq 2 $, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: $ u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w) $, $ z_t = \Delta z-\rho z+\mu u $, $ w_t = \Delta w-\delta w+\gamma u $, where $ \chi,\mu,\xi,\gamma,\rho,\delta>0 $, and the diffusivity $ D $ fulfills $ D(s)\geq c_0(s+1)^{M-1} $ for any $ s\geq 0 $ with $ c_0>0 $ and $ M\in\Bbb R $. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. $ \chi\mu = \xi\gamma $), the solution is globally bounded if $ M>-1 $ and $ n = 1 $, or $ M>2-4/n $ with $ n\geq 2 $. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.

keywords: Chemotaxis global boundedness quasilinear fully parabolic indirect signal production
Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations
Boling Guo Yan Lv Wei Wang
Discrete & Continuous Dynamical Systems - A 2014, 34(7): 2795-2818 doi: 10.3934/dcds.2014.34.2795
This paper derives a Schrödinger approximation for weakly dissipative stochastic Klein--Gordon--Schrödinger equations with a singular perturbation and scaled small noises on a bounded domain. Detail uniform estimates are given to pass the limit as perturbation and noise disappear. Approximation in two different spaces are considered. Furthermore a large deviation principe of solutions is derived by weak convergence approach.
keywords: large deviation principle weak convergence. Yukawa coupling Klein--Gordon--Schrödinger equations

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