Boundary behavior of large solutions to a class of Hessian equations

In this paper, we consider the m-Hessian equation S m [ D 2 u ] = b ( x ) f ( u ) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂ Ω. We give estimates of the asymptotic behavior of such solutions near ∂ Ω when the nonlinear term f satisfies a new structure condition.

Research on Stochastic Stability of Wheelsets with Primary Suspension

A new model for elastic constraint wheelset system of rail vehicle is proposed. Assuming the stochastic excitation as Gauss white noise, a stochastic model is built for elastic constraint wheelset system. Here two kinds of stochastic excitations are considered: one is the internal multiplicative excitation inherited in the internal system such as the spring and wheelset/rail contact geometric relationship, the other is the external excitation induced by track random irregularities. The model defined here is considered as a weak damping, weak excitation quasi non-integrable Hamiltonian system. The maximal Lyapunov exponent is calculated by quasi non-integrable Hamiltonian theory and oseledec multiplicative ergodic theory, and the stochastic local stability conditions are obtained. Meanwhile, the stochastic global stability conditions are derived by considering the modality of the singular boundary condition.

Stokes flow near the contact line of an evaporating drop

The evaporation of sessile drops in quiescent air is usually governed by vapour diffusion. For contact angles below $9{0}^{\ensuremath{\circ} } $, the evaporative flux from the droplet tends to diverge in the vicinity of the contact line. Therefore, the description of the flow inside an evaporating drop has remained a challenge. Here, we focus on the asymptotic behaviour near the pinned contact line, by analytically solving the Stokes equations in a wedge geometry of arbitrary contact angle. The flow field is described by similarity solutions, with exponents that match the singular boundary condition due to evaporation. We demonstrate that there are three contributions to the flow in a wedge: the evaporative flux, the downward motion of the liquid–air interface and the eigenmode solution which fulfils the homogeneous boundary conditions. Below a critical contact angle of $133. {4}^{\ensuremath{\circ} } $, the evaporative flux solution will dominate, while above this angle the eigenmode solution dominates. We demonstrate that for small contact angles, the velocity field is very accurately described by the lubrication approximation. For larger contact angles, the flow separates into regions where the flow is reversing towards the drop centre.

Quenching for a Non-Newtonian Filtration Equation with a Singular Boundary Condition

This paper deals with a nonlinearp-Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatisx=1are obtained. Moreover, the quenching rate of this equation is established. Finally, we give an example of an application of our results.

The solvability of an elliptic system under a singular boundary condition

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.