Related papers: The surface diffusion flow with elasticity in the …
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of…
In a previous work the authors described a fast high-fidelity computer model for acoustic scattering from multi-layered elastic spheres. This work is now extended with a scaling strategy significantly mitigating the problem of overflow and…
The traveling waves for surface diffusion of plane curves are studied. We consider an evolving plane curve with two endpoints, which can move freely on the x-axis with generating constant contact angles. For the evolution of this plane…
In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.
The flux-across-surfaces theorem (FAST) describes the outgoing asymptotics of the quantum flux density of a scattering state. The FAST has been proven for potential scattering under conditions on the outgoing asymptote $\psi_{\text{out}}$…
We obtained steady solutions to the two-dimensional Boussinesq approximation equations without mean temperature gradient. This system is referred to as free convection in this paper. Under an external flow described by the stream function…
We consider a surface diffusion flow of the form $V=\partial_s^2f(-\kappa)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $\kappa$ denotes the curvature and $V$…
We consider the linear stability of shear banded planar Couette flow of the Johnson-Segalman fluid, with and without the addition of stress diffusion to regularise the equations. In particular, we investigate the effect of two-dimensional…
A general reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady state solutions are proved via studying an equivalent…
We investigate the asymptotic stability of the length-penalized elastic flow of curves with boundary points constrained to the $x$-axis in $\mathbb{R}^2$. The main tool in our analysis is the Lojasiewicz--Simon inequality, which is used to…
We establish regularity and, under suitable assumptions, convergence to stationary states for weak solutions of a parabolic equation with a non-linear non-local drift term; this equation was derived from a model of active Brownian particles…
This paper is concerned with the existence and qualitative properties of pulsating fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. We focus especially on the influence of the spatial period and,…
We propose an approach to measure surface elastic constants of soft solids. Generally, this requires one to probe interfacial mechanics at around the elastocapillary length scale, which is typically microscopic. Deformations of microscopic…
In this paper we consider the entire weak solutions of the equations for stationary flows of shear thickening fluids in the plane and prove Liouville theorem under the global boundedness condition of velocity fields.
We show that certain radially symmetric steady states of compressible viscous fluids in domains with inflow/outflow boundary conditions are unconditionally stable. This means that any not necessarily radially symmetric solution of the…
We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is…
Drop condensation and evaportation as a result of the gradient in vapor concentration are important in both engineering and natural systems. One of the interesting natural examples is transpiration on plant leaves. Most of water in the…
Using extended Layzer's potential flow model, we investigate the effects of surface tension on the growth of the bubble and spike in combined Rayleigh-Taylor and Kelvin-Helmholtz instability. The nonlinear asymptotic solutions are obtained…
In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=\pm L$ moving relative to each other with opposite velocities $(\pm \alpha L,0,0)$ along the $x$-direction. Assuming that the…
We consider normal velocity of smooth sets evolving by the $s-$fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $s\in…