Asymptotics toward viscous contact waves for solutions of the Landau equation

In the paper, we are concerned with the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. Precisely, for the corresponding Cauchy problem in the spatially one-dimensional setting, we construct the unique global-in-time solution near a local Maxwellian whose fluid quantities are the viscous contact waves in the sense of hydrodynamics and also prove that the solution tends toward such local Maxwellian in large time. The result is proved by elaborate energy estimates and seems the first one on the dynamical stability of contact waves for the Landau equation. One key point of the proof is to introduce a new time-velocity weight function that includes an exponential factor of the form $\exp (q(t)\langle \xi\rangle^2)$ with $$q(t):=q_1-q_2\int_0^tq_3(s)\,ds,$$ where $q_1$ and $q_2$ are given positive constants and $q_3(\cdot)$ is defined by the energy dissipation rate of the solution itself. The time derivative of such weight function is able to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case. Note that in our problem the explicit time-decay of solutions around contact waves is unavailable but no longer needed under the crucial use of the above weight function, which is different from the situation in [11, 14].Comment: 60 pages. Comments are most welcome. Slight modifications have been made to simplify some estimates using the conservation of mas

KdV limit for the Vlasov-Poisson-Landau system

Two fundamental models in plasma physics are given by the Vlasov-Poisson-Landau system and the compressible Euler-Poisson system which both capture the complex dynamics of plasmas under the self-consistent electric field interactions at the kinetic and fluid levels, respectively. Although there have been extensive studies on the long wave limit of the Euler-Poisson system towards Korteweg-de Vries equations, few results on this topic are known for the Vlasov-Poisson-Landau system due to the complexity of the system and its underlying multiscale feature. In this article, we derive and justify the Korteweg-de Vries equations from the Vlasov-Poisson-Landau system modelling the motion of ions under the Maxwell-Boltzmann relation. Specifically, under the Gardner-Morikawa transformation $$(t,x,v)\to (\delta^{\frac{3}{2}}t,\delta^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t),v)$$ with $\varepsilon^{\frac{2}{3}}\leq \delta\leq \varepsilon^{\frac{2}{5}}$ and $\varepsilon>0$ being the Knudsen number, we construct smooth solutions of the rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the Korteweg-de Vries equations as $\delta\to 0$. Moreover, the explicit rate of convergence in $\delta$ is also obtained. The proof is obtained by an appropriately chosen scaling and the intricate weighted energy method through the micro-macro decomposition around local Maxwellians.Comment: 68 pages. All comments are welcome. arXiv admin note: text overlap with arXiv:2212.0765

Investing with Fast Thinking

Using data from a major online peer-to-peer lending market, we document that investors follow a simple rule of thumb under time pressure: they rush to invest in loans with high interest rates without sufficiently examining credit ratings, which are freely available on the trading interface. Our experiments show that making credit rating information more salient “nudges” investors into better decisions. Firsthand experience matters for learning for non-informational reasons: An investor responds differently when observing a default of her own loan, relative to observing a default of another investor’s loan

Characterization of Oil Droplets in the Upper Cavity of a Rotary Compressor

The lubricant oil is used to lubricate the bearings and seal the clearance between the sliding parts in the rotary compressor equipped in an air conditioning system. However, a portion of the oil is atomized and exhausted with the refrigerant flow, which reduces the system’s efficiency and reliability. In this study, a highresolution shadowgraph system is applied to a modified rotary compressor to visualize the oil droplets in its upper cavity. The oil droplet size distribution is studied statistically at different radial locations and different crankshaft’s frequencies. It is observed that more oil droplets with larger D10 and D32 are generated as the rotating frequency increases. There are fewer droplets moving toward the center of the compressor. More oil droplets in the upper cavity at higher frequency might increase the Oil Discharge Rate (ODR) if these droplets are carried into the downstream discharge tube. On the other hand, larger droplets at a higher frequency may reduce ODR since the appropriate aerodynamic design may retain these droplets in the upper cavity without moving into the discharge tube. The result can assist designers in improving the CFD analysis of compressors and ultimately reducing ODR

Transferable E(3) equivariant parameterization for Hamiltonian of molecules and solids

Using the message-passing mechanism in machine learning (ML) instead of self-consistent iterations to directly build the mapping from structures to electronic Hamiltonian matrices will greatly improve the efficiency of density functional theory (DFT) calculations. In this work, we proposed a general analytic Hamiltonian representation in an E(3) equivariant framework, which can fit the ab initio Hamiltonian of molecules and solids by a complete data-driven method and are equivariant under rotation, space inversion, and time reversal operations. Our model reached state-of-the-art precision in the benchmark test and accurately predicted the electronic Hamiltonian matrices and related properties of various periodic and aperiodic systems, showing high transferability and generalization ability. This framework provides a general transferable model that can be used to accelerate the electronic structure calculations on different large systems with the same network weights trained on small structures.Comment: 33 pages, 6 figure