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This paper is concerned with the following L 2 -subcritical Kirchhoff-type equation − a + b ∫ R 2 | ∇ u | 2 d x s Δ u + V (x) u = μ u + β | u | 2 u , x ∈ R 2 , with ∫ R 2 | u | 2 d x = 1 . We give a detailed analysis of the limit property of the L 2 -normalized solution when exponent s tends toward 0 from the right (i.e., s ↘ 0 ). Our research extends previous works, in which the authors have displayed the limit behavior of L 2 -normalized solutions when s = 1 as a ↘ 0 or b ↘ 0 . [ABSTRACT FROM AUTHOR]

Accurately determining the seismic structure of the continental deep crust is crucial for understanding its geological evolution and continental dynamics in general. However, traditional tools such as surface waves often face challenges in solving the trade‐offs between elastic parameters and discontinuities. In this work, we present a new approach that combines two established inversion techniques, receiver function H‐κ stacking and joint inversion of surface wave dispersion and receiver function waveforms, within a Bayesian Monte Carlo (MC) framework to address these challenges. Demonstrated by synthetic tests, the new method greatly reduces trade‐offs between critical parameters, such as the deep crustal Vs, Moho depth, and crustal Vp/Vs ratio. This eliminates the need for assumptions regarding crustal Vp/Vs ratios in joint inversion, leading to a more accurate outcome. Furthermore, it improves the precision of the upper mantle velocity structure by reducing its trade‐off with Moho depth. Additional notes on the sources of bias in the results are also included. Application of the new approach to USArray stations in the Northwestern US reveals consistency with previous studies and identifies new features. Notably, we find elevated Vp/Vs ratios in the crystalline crust of regions such as coastal Oregon, suggesting potential mafic composition or fluid presence. Shallower Moho depth in the Basin and Range indicates reduced crustal support to the elevation. The uppermost mantle Vs, averaging 5 km below Moho, aligns well with the Pn‐derived Moho temperature variations, offering the potential of using Vs as an additional constraint to Moho temperature and crustal thermal properties. Plain Language Summary: Knowing the seismic structure of the deep crust helps us understand Earth's geological history and how continents evolve. However, traditional methods of studying the deep crust face challenges due to tradeoffs that can impact accuracies of the results. In this paper, we present a new approach that combines two existing techniques intending to measure the deep crust more accurately. We tested this method using both synthetic and real data and learned that it works better than previous methods. We applied this method to the Northwestern US and found that the results are aligned with the area's geology, suggesting that the new method is feasible to be applied on a regional scale. The new method provides a more accurate way to study the deep crust and improves the mapping of the uppermost mantle. Key Points: We have developed a new joint inversion approach that incorporates stacking of receiver function multiple phases with multiple data setsThe new approach reduces the trade‐offs and improves the determination of deep crustal shear velocity, Moho, and Poisson's ratioApplication of the new method to the northwestern US produces a more accurate model that exhibits geologically coherent structures [ABSTRACT FROM AUTHOR]

Hydrophobicity has been developed in many areas, whose potentials in drag reduction at microscale have attracted numerous attentions for expanding the practical applications in fields of on chip devices, materials synthesis, and enhanced heat transfer. In this article, we select polydimethylsiloxane (PDMS) as the base material, whose hydrophobic modifications have been well developed. Among them, hydrofluoric acid treated one shows great performance and leads us to two types of microchannels, the straight and U-shaped, with enhanced hydrophobicity (from 91° to 106°). The coefficients of the pressure drop are experimentally measured with the Reynolds number ranging from 0 to 300. The results illustrate that the drag reduction rate reaches at 37.8% for the straight microchannel and 26.8% for the U-shaped microchannel. With the increase in the Reynolds number, the drag reduction effect stays almost constant for the straight channel, while it decreases gradually for the U-shaped channel. The flow impingement induced by a centrifugal force has an important impact on the slip effect that grows with the Re. Next, we adopt the numerical method and the micro-particle imaging velocimetry measurement to analyze the drag reduction effect from perspectives of the slip length. We successfully derive the slip length model correlating the drag reduction effect. Our results not only achieve substantial drag reduction in PDMS microchannels, but also provide a quantitative correlation between hydrophobicity and drag reduction, offering a feasible strategy for extensive applications at microscale, such as fluid actuation, bio-chip analysis, and highly efficient cooling system. [ABSTRACT FROM AUTHOR]

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For s = 1 , there many articles have analyzed the limit behavior of minimizers when η > 0 as b → 0 + or b > 0 as η → 0 + . When the equation involves a varying non-local term ∫ R 3 | ∇ u | 2 d x s , we give a detailed limit behavior analysis of constrained minimizers for any positive sequence { η k } with η k → 0 + . The present paper obtains an interesting result on this topic and enriches the conclusions of previous works. [ABSTRACT FROM AUTHOR]

Flowing characteristics of drag reduction are experimentally and numerically investigated with a combination of hydrophobic coating and various cavities in polydimethylsiloxane microchannels. Two typical types of cavities with different geometrical parameters, rectangular and triangular, are studied with a wide range of Reynolds numbers from 0 to 300. A promising hydrophobic processing method is proposed based on a comprehensive performance on surface morphology, water repellency, and structural distortion. In this method, hydrofluoric acid (40 wt. %) is adopted and the corrosion time is set to 5 min. The velocity field and streamlines are acquired by the micro-particle image velocimetry system and numerical models to explain the flow patterns in particular. The slip length is measured as 13.38 μm in the hydrophobic straight channel. For rectangular cavities, the drag reduction rate reaches nearly 14.1% under no-slip condition and 33.2% under slip condition. A critical turning point of the co-effect is found by numerical results when the slip length is about 15 μm, which is also determined by the cavities. The convergent and divergent angles of triangular cavities play a critical role in the pressure drop due to the competition of the vortex and flow impingement. A nonlinear model is developed based on the numerical results to predict Poiseuille number with the relevant important variables for a two-dimensional microchannel. Our results reveal the fundamental physics of flowing characteristics with the co-design of hydrophobicity and microstructures, predicting a composite design method for widespread applications in microfluids. [ABSTRACT FROM AUTHOR]