On the local in time well-posedness of an elliptic–parabolic ferroelectric phase-field model

We consider a state-of-the-art ferroelectric phase-field model arising from the engineering area in recent years, which is mathematically formulated as a coupled elliptic–parabolic differential system. We utilize the maximal parabolic regularity theory to show the local in time well-posedness of the ferroelectric problem in both 2D and 3D spaces, which is sharp in the sense that the local solution is unique and a blow-up criterion is present. The well-posedness result will firstly be proved under some general assumptions. Afterwards we give sufficient geometric and regularity conditions which will guarantee the fulfillment of the imposed assumptions.     

Remarks on Liouville Type Result for the 3D Hall-MHD System

In this paper, we consider the 3D Hall-MHD system, and provide an improved Liouville type result for its stationary version.     

Global well‐posedness of the nonhomogeneous incompressible liquid crystals systems

This paper examines the initial‐value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow‐up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.     

On(e)-reducible 3-manifolds Which Admit Complete Surface Systems

In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M , then there exists a complete surface system S for M , which is equivalent to S′, such that D is disjoint from S . We also obtain some properties of such 3-manifolds which can be embedded in S3.     

On the nonlocal stabilization by starting control of the normal equation generated from the Helmholtz system

We consider the problem of stabilization near zero of semilinear normal parabolic equations connected with the 3D Helmholtz system with periodic boundary conditions and arbitrary initial datum. This problem was previously studied in Fursikov and Shatina (2018). As it was recently revealed, the control function suggested in that work contains a term impeding transferring the stabilization construction on the 3D Helmholtz system. The main concern of this paper is to prove that this term is not necessary for the stabilization result, and therefore the control function can be changed by a proper way.     

Global well-posedness and scattering for a nonlinear Klein–Gordon system in low dimensions

This article investigates the global well-posedness and the scattering for a nonlinear Klein–Gordon system in spatial dimensions 1 and 2. We establish a Morawetz estimate for this system which is similar to the Morawetz estimate established by Nakanishi [K. Nakanishi, Energy scattering for nonlinear Klein–Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169(1), pp. 201–225], combining this Morawetz estimate with the induction on energy argument developed by Bourgain [J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Am. Math. Soc. 12 (1999), pp. 145–171], the bound of a certain space-time norm and scattering result are obtained.     

Global Dynamics below the Ground State Energy for the Klein-Gordon-Zakharov System in the 3D Radial Case

We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up.     

On the Non-Existence of the Closed Orbit for a Liénard System

A sufficient condition for a Liénard system to have no non-trivial closed orbits is given by transforming the system into another system called the Bogdanov—Takens system. The result here (Theorem 2) is a partial improvement of that of Wang and Yu [2].AMS Subject Classification (2000), 34C07, 34C25, 34C26, 34D20     

Extended Center Manifold, Global Bifurcation and Approximate Solutions of Chen chaotic dynamical system

The study of the chaotic Chen dynamic System (CDS) has been a recent focus in the literature, with numerous works exploring its various chaotic features. However, the majority of these studies have relied primarily on numerical techniques to investigate nonlinear dynamic systems (NLDSs). In this context, our aim is to derive approximate analytical solutions for the CDS by developing an iterative scheme. We have proven the convergence theorem for this scheme, which ensures that our iterative process will converge to the exact solution. Additionally, we introduce a new method for constructing the extended center manifold, a critical component in the analysis of dynamical systems. The characteristics of the global bifurcation of the system components within the parameter space are explored. The error analysis of the iterated solutions demonstrates the efficiency of the present technique. We present both three-dimensional (3D) and two-dimensional (2D) phase portraits of the system. The 3D portrait reveals a feedback loop pattern, while the 2D portrait, which represents the interaction of the system components, exhibits multiple pools and cross pools. Furthermore, we illustrate the global bifurcation by visualizing the components of the CDS against the space parameters. The sensitivity of CDS to innitesimal variations in the initial conditions (ICs) are tested. It is found that even minor changes can lead to signicant alterations in the system.     

A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows

In this paper, we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplified 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal. The result shows that if a local smooth solution (u,d) satisfies $$∫^T_0\frac{||u||^{\frac{2}{1-r}}_{\dot{B}^{-r}{∞,∞}}+||∇ d||²_{L^∞}}{1+1n(e+||u||_H^S+||∇ d||_H^S)}dt‹∞$$ with 0 ≤ r ‹ 1 and s ≥ 3, then the solution (u,d) can be smoothly extended beyond the time T.     

HOPF BIFURCATION ANALYSIS OF A 3D CHAOTIC SYSTEM

In this paper,a 3D chaotic system with multi-parameters is introduced. The dynamical systems of the original ADVP circuit and the modified ADVP model are regarded as special examples to the system.Some basic dynamical behaviors such as the stability of equilibria,the existence of Hopf bifurcation are investigated.We analyse the Hopf bifurcation of the system comprehensively using the first Lyapunov coefficient by precise symbolic computation.As a result,in fact we have studied the further dynamical behaviors.     

A Steiner triple system which colors all cubic graphs

We prove that there is a Steiner triple system ?? such that every simple cubic graph can have its edges colored by points of ?? in such a way that for each vertex the colors of the three incident edges form a triple in ??. This result complements the result of Holroyd and ?koviera that every bridgeless cubic graph admits a similar coloring by any Steiner triple system of order greater than 3. The Steiner triple system employed in our proof has order 381 and is probably not the smallest possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 15–24, 2004     

A model for lime consolidation of porous solids

We propose a mathematical model describing the process of filling the pores of a building material with lime water solution with the goal to improve the consistency of the porous solid. Chemical reactions produce calcium carbonate which glues the solid particles together at some distance from the boundary and strengthens the whole structure. The model consists of a 3D convection–diffusion system with a nonlinear boundary condition for the liquid and for calcium hydroxide, coupled with the mass balance equations for the chemical reaction. The main result consists in proving that the system has a solution for each initial data from a physically relevant class. A 1D numerical test shows a qualitative agreement with experimental observations.     

The existence of random $\mathcal{D}$-pullback attractors for random dynamical system and its applications

In this paper, we establish a result on the existence of random $\mathcal{D}$-pullback attractors for norm-to-weak continuous non-autonomous random dynamical system. Then we give a method to prove the existence of random $\mathcal{D}$-pullback attractors. As an application, we prove that the non-autonomous stochastic reaction diffusion equation possesses a random $\mathcal{D}$-pullback attractor in $H_0^1$ with polynomial growth of the nonlinear term.     

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.      

Exact controllability in projections for three-dimensional Navier–Stokes equations

The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Our sufficient condition is verified for any torus in R³${\mathbb{R}}^3$. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier–Stokes system has a unique strong solution for any initial function and a large class of external forces.     

Weak solutions to stochastic 3D Navier-Stokes-α model of turbulence: α-Asymptotic behavior

We study the asymptotic behavior of weak solutions to the stochastic 3D Navier-Stokes-α model as α approaches zero. The main result provides a new construction of the weak solutions of stochastic 3D Navier-Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier-Stokes-α model.     

Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3-10].     

Spectrum comparison for a conserved reaction-diffusion system with a variational property

We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010