Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

Controlling Chaotic Vibrations of a Quarter-Car Model Excited by Road Surface Profile using Parametric Excitation.

Chaotic Vibrations are considered for a quarter-car model excited by the road surface profile. The equation of motion is obtained in the form of a classical Duffing equation and it is modeled with deliberate introduction of parametric excitation force term to enable us manipulate the behavior of the system. The equation of motion is solved using the Method of Multiple Scales. The steady-state solutions with and without the parametric excitation force term is investigated using NDSolve MathematicaTM Code and the nonlinear dynamical system’s analysis is by a study of the Bifurcations that are observed from the analysis of the trajectories, and the calculation of the Lyapunov. In making the system more strongly nonlinear the excitation amplitude value is artificially increased to various multiples of the actual value. Results show that the system’s response can be extremely sensitive to changes in the amplitude and the that chaos is evident as the system is made more nonlinear and that with the introduction of parametric excitation force term the system’s motion becomes periodic resulting in the elimination of chaos and the reduction in amplitude of vibration.

Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

Existence of strong solutions to the rotating shallow water equations with degenerate viscosities

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

Stability of an Axisymmetric Liquid Metal Flow Driven by a Multi-Pole Rotating Magnetic Field

The stability of an electrically conducting fluid flow in a cylinder driven by a multi-pole rotating magnetic field is numerically studied. A time-averaged Lorentz force term including the electric potential is derived on the condition that the skin effect can be neglected and then it is incorporated into the Navier-Stokes equation as a body force term. The axisymmetric velocity profile of the basic flow for the case of an infinitely long cylinder depends on the number of pole-pairs and the Hartmann number. A set of linearized disturbance equations to obtain a neutral state was successfully solved using the highly simplified marker and cell (HSMAC) method together with a Newton–Raphson method. For various cases of the basic flow, depending on both the number of pole-pairs and the Hartmann number, the corresponding critical rotational Reynolds numbers for the onset of secondary flow were obtained instead of using the conventional magnetic Taylor number. The linear stability analyses reveal that the critical Reynolds number takes its minimum at a certain value of the Hartmann number. On the other hand, the velocity profile for cases of a finite length cylinder having a no-slip condition at the flat walls generates the Bödewadt boundary layers and such flows need to be computed including the non-linear terms of the Navier-Stokes equation.

Kinetic theory-based force analysis in lattice Boltzmann equation

In the gas kinetic theory, it is shown that the zeroth order of the density distribution function [Formula: see text] and local equilibrium density distribution function were the Maxwellian distribution [Formula: see text] with an external force term, where [Formula: see text] is the fluid density, [Formula: see text] is the physical velocity and [Formula: see text] is the temperature, while in the lattice Boltzmann equation (LBE) method, numerous force treatments were proposed with a discrete density distribution function [Formula: see text] apparently relaxed to a given state [Formula: see text], where the given velocity [Formula: see text] could be different with [Formula: see text], and the Chapman–Enskog (CE) analysis showed that [Formula: see text] and local equilibrium density distribution function should be [Formula: see text] in the literature. In this paper, we start from the kinetic theory and show that the [Formula: see text] and local equilibrium density distribution function in LBE should obey the Maxwellian distribution [Formula: see text] with [Formula: see text] relaxed to [Formula: see text], which are consistent with kinetic theory. Then the general requirements for the force term are derived, by which the correct hydrodynamic equations could be recovered at Navier–Stokes (NS) level, and numerical results confirm our theoretical analysis.