Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad \partial_t u + \partial_x p(v) =\displaystyle -\frac{\alpha}{(1+t)^\lambda} u, \end{equation} with null-Dirichlet boundary condition or null-Neumann boundary condition on $u$. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang (1999, J. Differential Equations, 156, 439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the facts that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave

Exciton Binding Energy of Monolayer WS2

The optical properties of monolayer transition metal dichalcogenides (TMDC) feature prominent excitonic natures. Here we report an experimental approach toward measuring the exciton binding energy of monolayer WS2 with linear differential transmission spectroscopy and two-photon photoluminescence excitation spectroscopy (TP-PLE). TP-PLE measurements show the exciton binding energy of 0.71eV around K valley in the Brillouin zone. The trion binding energy of 34meV, two-photon absorption cross section 4X10^{4}cm^{2}W^{-2}S^{-1} at 780nm and exciton-exciton annihilation rate around 0.5cm^{2}/s are experimentally obtained.Comment: 5page,3 figure