On the discretisation in time of the stochastic Allen–Cahn equation

We consider the stochastic Allen–Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension , and study the semidiscretisation in time of the equation by an Euler type split‐step method with step size . We show that the method converges strongly with a rate . By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.     

Existence of global strong solution for the compressible Navier–Stokes equations with degenerate viscosity coefficients in 1D

We consider Navier–Stokes equations for compressible viscous fluids in the one‐dimensional case. We prove the existence of global strong solution with large initial data for compressible Navier–Stokes equation with viscosity coefficients of the form with (it includes in particular the important physical case of the viscous shallow water system when ). The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity v (see 13 , 14 , 16 , 17 ) such that the density verifies a parabolic equation. We estimate v in norm which enables us to control the norm of by using the maximum principle.     

Time-varying many-server finite-queues in tandem: Comparing blocking mechanisms via fluid models

This paper focuses on the mechanism of Blocking Before Service (BBS), in time-varying many-server queues in tandem. BBS arises in telecommunication networks, production lines and healthcare systems. We model a stochastic tandem network under BBS and develop its corresponding fluid limit, which includes reflection due to jobs lost. Comparing our fluid model against simulation shows that the model is accurate and effective. This gives rise to design/operational insights regarding network throughput, under both BBS and BAS (Blocking After Service).     

Nonlinear iteration method for proximal split feasibility problems

The purpose of this paper is to introduce iterative algorithm which is a combination of hybrid viscosity approximation method and the hybrid steepest‐descent method for solving proximal split feasibility problems and obtain the strong convergence of the sequences generated by the iterative scheme under certain weaker conditions in Hilbert spaces. Our results improve many recent results on the topic in the literature. Several numerical experiments are presented to illustrate the effectiveness of our proposed algorithm, and these numerical results show that our result is computationally easier and faster than previously known results on proximal split feasibility problem.     

Asymptotic approximation for the solution to a semilinear parabolic problem in a thin star‐shaped junction

A semilinear parabolic problem is considered in a thin 3‐D star‐shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter The purpose is to study the asymptotic behavior of the solution u_ε as ε→0, ie, when the star‐shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors and in nonlinear perturbed Robin boundary conditions. We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {α_i}and {β_i}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε→0. Namely, in each case, we derive the limit problem (ε=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

Strong comparison principles for some nonlinear degenerate elliptic equations

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ▽~2ψ + L(x, ▽ψ), including the conformal hessian operator.     

Equimatchable claw-free graphs

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm.     

Rainbow number of matchings in planar graphs

The rainbow number$rb(G,H)$ for the graph $H$ in $G$ is defined to be the minimum integer $c$ such that any $c$-edge-coloring of $G$ contains a rainbow $H$. As one of the most important structures in graphs, the rainbow number of matchings has drawn much attention and has been extensively studied. Jendrol et al. initiated the rainbow number of matchings in planar graphs and they obtained bounds for the rainbow number of the matching $k{K}_2$ in the plane triangulations, where the gap between the lower and upper bounds is $O\left(k^3\right)$. In this paper, we show that the rainbow number of the matching $k{K}_2$ in maximal outerplanar graphs of order $n$ is $n+O\left(k\right)$. Using this technique, we show that the rainbow number of the matching $k{K}_2$ in some subfamilies of plane triangulations of order $n$ is $2n+O\left(k\right)$. The gaps between our lower and upper bounds are only $O\left(k\right)$.