Riesz potential estimates for a class of double phase problems

A double phase problem with a bounded Borel measure on the right hand side is studied. We prove an optimal pointwise gradient estimate for such a measure data problem via Riesz potentials of the measure under log-Dini continuity assumption on the modulating coefficients. As a consequence, we find an optimized $C^1$ regularity criterion.     

Remarks on the asymptotic behavior of scalar auxiliary variable (SAV) schemes for gradient-like flows

We introduce a time semi-discretization of a damped wave equation by a SAV scheme with second order accuracy. The energy dissipation law is shown to hold without any restriction on the time step. We prove that any sequence generated by the scheme converges to a steady state (up to a subsequence). We notice that the steady state equation associated to the SAV scheme is a modified version of the steady state equation associated to the damped wave equation. We show that a similar result holds for a SAV fully discrete version of the Cahn-Hilliard equation and we compare numerically the two steady state equations.     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

Gradient based and least squares based iterative algorithms for matrix equations AXB + CXD = F

This paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation AXB + CX^TD = F. The basic idea is to decompose the matrix equation (system) under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving Ax = b and AXB = F. The analysis shows that when the matrix equation has a unique solution (under the sense of least squares), the iterative solution converges to the exact solution for any initial values. A numerical example verifies the proposed theorems.     

Gradient Estimates for the Heat Kernels in Higher Dimensional Heisenberg Groups

The author obtains sharp gradient estimates for the heat kernels in two kinds of higher dimensional Heisenberg groups -- the non-isotropic Heisenberg group and the Heisenberg type group Hn,m. The method used here relies on the positive property of the Bakry-Emery curvature F2 on the radial functions and some associated semigroup technics.     

Simulation of pedestrian counter flow through bottlenecks by using an agent-based model

Considerable research has been conducted on the topic of unidirectional evacuations from exits. However, few studies aim at simulating counter flow through a bottleneck with complex conflict. This paper proposes an agent-based model to investigate bidirectional flow evacuation. Pedestrian speed is determined by the speed of the leading agent and the surrounding agents. The moving direction of pedestrian originates from four forces, namely, gradient force, repulsive force, resistance force, and random force. These four forces dominate the main stream of the pedestrian moving trajectory, the interaction between pedestrians and their local environment, the resistance or disinclination to movement, and the random variations and chaotic nature of pedestrian dynamics. The novelty of this research is in the agent-based model that combines the agent and forces while providing insights for the simulation of the pedestrian dynamic on the cognitive level. The experiment results show that the behavior that arises from this model is consistent with the observations from Guangzhou Metro and that this model could help capture the essence of pedestrian behavior near egresses.     

On a Nonlinear,Nonlocal Parabolic Problem with Conservation of Mass,Mean and Variance

In this paper we prove that the steepest descent of certain porous-medium type functionals with respect to the quadratic Wasserstein distance over a constrained (but not weakly closed) manifold gives rise to a nonlinear, nonlocal parabolic partial differential equation connected to the study of the asymptotic behavior of solutions for filtration problems. The result by Carlen and Gangbo on constrained optimization for steepest descent of the negative Boltzmann entropy in the Wasserstein space is generalized to porous-medium type functionals. An interesting feature of the resulting Fokker-Planck equation is the nonlocality of its drift term occurring at the same time as its nonlinearity.     

Shear banding in semidilute entangled polymer solutions

Shear banding in semidilute polymer solutions and other soft materials is one of the most intensely debated topics in current rheology. By means of rheo-optical experiments, physical modeling, and numerical simulations, researchers have started to develop a more thorough understanding of this flow instability within the past few years. Nevertheless, much effort is still required to identify the exact microscopic mechanisms leading to shear band formation and to clarify whether the phenomenon is universal for polymers. For this purpose, basic rheological characteristics, such as the appearance of a stress overshoot during start-up of a simple shear flow, have to be revisited and better related to the dynamics of the polymeric network.     

Gradient force evolution of Gaussian beam induced by a phase plate

The evolution of gradient force pattern induced by an annular phase distribution plate is numerically investigated in this paper. The phase plate, which may alter the wavefront phase of incident Gaussian beam with tunable topological charge, consists of two concentric portions, one center circle portion and one annular portion. Numerical simulations show that the proposed plate can induce the tunable gradient force on the particles in the focal region. By adjusting the geometrical parameters or changing the topological charge of the phase-shifting plate, some novel trap patterns may occur, such as triangle shape trap, quadrangle shape trap, pentagon shape trap, hexagon shape trap, and the shapes of optical traps change very considerably. Therefore, the phase plate may be very advantageous for constructing tunable optical traps. The method is more versatile in that it allows precise control of the parameters and has the possibility of generating specific patterns of optical vortices. The gradient force pattern focal of intensity distribution depends on both the annular width and the topological charge.