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We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $\mathbb{T}\times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved. 

Circular microdisk mechanical resonators vibrating in their various resonance modes have emerged as important platforms for a wide spectrum of technologies including photonics, cavity optomechanics, optical metrology, and quantum optics. Optically transduced microdisk resonators made of advanced materials such as silicon carbide (SiC), diamond, and other wide- or ultrawide-bandgap materials are especially attractive. They are also of strong interest in the exploration of transducers or detectors for harsh environments and mission-oriented applications. Here we report on the first experimental investigation and analysis of energetic proton radiation effects on microdisk resonators made of 3C-SiC thin film grown on silicon substrate. We fabricate and study microdisks with diameters of ∼48 µm and ∼36 µm, and with multimode resonances in the ∼1 to 20 MHz range. We observe consistent downshifts of multimode resonance frequencies, and measure fractional frequency downshifts from the first three flexural resonance modes, up to ∼-3420 and -1660 ppm for two devices, respectively, in response to 1.8 MeV proton radiation at a dosage of 10¹⁴/cm². Such frequency changes are attributed to the radiation-induced Young’s modulus change of ∼0.38% and ∼0.09%, respectively. These devices also exhibit proton detection responsivity of ℜ ≈ -5 to -6 × 10⁻⁶Hz/proton. The results provide new knowledge of proton radiation effects in SiC materials, and may lead to better understanding and exploitation of micro/nanoscale devices for harsh-environment sensing, optomechanics, and integrated photonics applications.

 

We prove asymptotic stability of point vortex solutions to the full Euler equation in two dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as $t\to\infty$ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping. © 2021 Wiley Periodicals LLC.

 

While human scleral and corneal tissues possess similar structural morphology of long parallel cylindrical collagen fibrils, their optical characteristics are markedly different. Using pseudospectral time-domain (PSTD) simulations of Maxwell’s equations, we model light propagation through realistic representations of scleral and corneal nanoarchitecture and analyze the transmittance and spatial correlation in the near field. Our simulation results provide differing predictions for scleral opacity and corneal transparency across the vacuum ultraviolet to the mid-infrared spectral region in agreement with experimental data. The simulations reveal that the differences in optical transparency between these tissues arise through differences in light scattering emanating from the specific nanoscale arrangement and polydispersity of the constituent collagen fibrils.