Abstract. We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hörmander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.

The abundance of low-temperature waste heat produced by industry and automobile exhaust necessitates the development of power generation with thermoelectric (TE) materials. Commercially available bismuth telluride-based alloys are generally used near room temperature. Materials that are composed of p-type bismuth telluride, which are suitable for low-temperature power generation (near 380 K), were successfully obtained through Sb-alloying, which suppresses detrimental intrinsic conduction at elevated temperatures by increasing hole concentrations and material band gaps. Furthermore, hot deformation (HD)-induced multi-scale microstructures were successfully realized in the high-performance p-type TE materials. Enhanced textures and donor-like effects all contributed to improved electrical transport properties. Multiple phonon scattering centers, including local nanostructures induced by dynamic recrystallization and high-density lattice defects, significantly reduced the lattice thermal conductivity. These combined effects resulted in observable improvement of ZT over the entire temperature range, with all TE parameters measured along the in-plane direction. The maximum ZT of 1.3 for the hot-deformed Bi 0.3 Sb 1.7 Te 3 alloy was reached at 380 K, whereas the average ZT av of 1.18 was found in the range of 300-480 K, indicating potential for application in low-temperature TE power generation. Keywords: bismuth telluride; donor-like effect; hot deformation; low-temperature power generation; texture INTRODUCTION Thermoelectric (TE) devices have attracted extensive interest over the past few decades because of their potential use in direct thermal-toelectrical energy conversion and solid-state refrigeration. The TE conversion efficiency of a material can be gauged by the dimensionless figure of merit ZT ¼ a 2 sT/k, where a, s, k and T are the Seebeck coefficient, the electrical conductivity, the thermal conductivity and the operating temperature, respectively. 1 Continuous effort has been invested toward improving the ZT values of TE materials, resulting in significant advances through phonon engineering 2-9 and band engineering. [10][11][12][13][14] For example, remarkable increases in ZT have been achieved in bulk nanomaterials via the enhancement of phonon scattering at boundaries to reduce lattice thermal conductivities. 2,4,6,7 Currently, the best commercial TE materials near room temperature are still rhombohedral bismuth tellurides and related solid solutions fabricated by unidirectional crystal growth. [15][16][17] Nanostructuring strategies have been devised to prepare highperformance bismuth telluride-based alloys, including bottom-up

Broadband photodetectors based on TiO 2 nanotubes (NTs) array have significant prospects in many fields such as environmental monitoring. Herein, a simple spin-coating process is successfully adopted to incorporate MAPbI 3 quantum dots (QDs) onto the surface of TiO 2 NTs to form a heterostructure, extending the response range of TiO 2 NT from ultraviolet to visible. Compared with pure TiO 2 NTs, the heterostructure demonstrates an improvement of responsivity in visible range by three orders of magnitude, and maintains its response performance in the UV range simultaneously. The TiO 2 NTs based heterostructure photodetectors demonstrate a relative fast and stable response in the 300-800 nm range and even have a reponsivity of 0.2 A W −1 at 700 nm. The photoelectric performance of the hybrid photodetector based on TiO 2 NTs maintains well when exposed to moist air for 72 h or heated from room temperature to 100 °C. Moreover, such a TiO 2 NTs/MAPbI 3 QDs heterostructure device demonstrates excellent flexibility and high transparency (85%) in the 400-800 nm range, their photodetecting performance is well retained after 200 cycles of repeated bending at 90°. The present strategy that combines facile electrospinning and solution-processed QDs may open a new avenue for wide range response and flexible devices construction.The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adfm.201703115.superior photoelectric performance for photovoltaic [9] and photocatalytic [10] applications resulted from improved charge collection efficiency by promoting carrier separation and charge transport. [11][12][13] Recently, TiO 2 NTs array is applied for photodetecting applications primarily due to its high surface-to-volume ratios, which can increase the contact area with oxygen, and the controllable properties by varying dimensions of NTs. [14] However, as a wide band gap (anatase, 3.2 eV; rutile, 3.0 eV) semiconductor, anatase TiO 2 NTs only have absorption in the UV region if directly applied, which limits their photodetecting application in the wide range. [15] Many efforts have been thus devoted in the chemical doping and sensitization of TiO 2 in the past for its sensitivity in the visible spectra. Doping of TiO 2 with carbon, nitrogen, or transition metals tends to reduce the band gap of TiO 2 , whereas the 1D morphologies of resulted nanostructures are hardly maintained. [16] Alternatively filling the NTs with organic dye or polymer, which have a wide absorption range. [17] Nonetheless, the process for doping TiO 2 NTs could not be well controlled, [18] and the process of filling organic into the NTs is pretty complex and the thermal stability is poor for organics. Therefore, achieving wide range photodetecting over an inherent energy band gap of NTs without inducing fatal degradation of other photoelectronic performance is still a challenge. [19] Perovskite (CH 3 NH 3 PbI 3 , MAPbI 3 ) quantum dots (QDs), with the merits of size tunable band gap (from 3.1 to 1.7 eV), hi...

Compressible vortex sheets are fundamental waves, along with shock and rarefaction waves, in entropy solutions to the multidimensional hyperbolic systems of conservation laws; and understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number M > √ 2 and instability when M < √ 2; and Artola-Majda's analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when M > √ 2. For the Euler equations in three-dimensions, every compressible vortex sheet is violently unstable and this violent instability is the analogue of the Kelvin-Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss-Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients without loss of the order. Then we exploit these results to develop a suitable iteration scheme of Nash-Moser-Hörmander type and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in the three-dimensional MHD.

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