A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary

We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.Comment: Final version, to appear in Nonlinearit

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

We consider 3D compressible isentropic Euler equations describing the motion of a liquid in an unbounded initial domain with a moving boundary and a fixed flat bottom at finite depth. The liquid is under the influence of gravity and surface tension and it is not assumed to be irrotational. We prove the local well-posedness by introducing carefully-designed approximate equations which are asymptotically consistent with the a priori energy estimates. The energy estimates yield no regularity loss and are uniform in Mach number. Also, they are uniform in surface tension coefficient if the Rayleigh-Taylor sign condition holds initially. We can thus simultaneously obtain incompressible and vanishing-surface-tension limits. The method developed in this paper is a unified and robust hyperbolic approach to free-boundary problems in compressible Euler equations. It can be applied to some important complex fluid models as it relies on neither parabolic regularization nor irrotational assumption. This paper joined with our previous works [46,47] rigorously proves the local well-posedness and the incompressible limit for a compressible gravity water wave with or without surface tension.Comment: 63 page

Calculation of the current noise spectrum in mesoscopic transport: an efficient quantum master equation approach

Based on our recent work on quantum transport [Li et al., Phys. Rev. B 71, 205304 (2005)], where the calculation of transport current by means of quantum master equation was presented, in this paper we show how an efficient calculation can be performed for the transport noise spectrum. Compared to the longstanding classical rate equation or the recently proposed quantum trajectory method, the approach presented in this paper combines their respective advantages, i.e., it enables us to tackle both the many-body Coulomb interactionand quantum coherence on equal footing and under a wide range of setup circumstances. The practical performance and advantages are illustrated by a number of examples, where besides the known results and new insights obtained in a transparent manner, we find that this alternative approach is much simpler than other well-known full quantum mechanical methods such as the Landauer-B\"uttiker scattering matrix theory and the nonequilibrium Green's function technique.Comment: 13 pages, 3 figures, submitted to PR

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh-Taylor sign condition holds true initially. This result is a continuation of the authors' previous works [17,32,16]. Our proof is based on the combination of the techniques developed in our previous works [17,32,16], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.Comment: 35 pages. Final version, accepted by Nonlinearity. We extend the result to the case of a general diffeomorphis

Full counting statistics of renormalized dynamics in open quantum transport system

The internal dynamics of a double quantum dot system is renormalized due to coupling respectively with transport electrodes and a dissipative heat bath. Their essential differences are identified unambiguously in the context of full counting statistics. The electrode coupling caused level detuning renormalization gives rise to a fast-to-slow transport mechanism, which is not resolved at all in the average current, but revealed uniquely by pronounced super-Poissonian shot noise and skewness. The heat bath coupling introduces an interdot coupling renormalization, which results in asymmetric Fano factor and an intriguing change of line shape in the skewness.Comment: 9 pages, 5 figure

Distance-dependent emission spectrum from two qubits in a strong-coupling regime

We study the emission spectrum of two distant qubits strongly coupled to a waveguide by using the numerical and analytical approaches, which are beyond the Markovian approximation and the rotating-wave approximation (RWA). The numerical approach combines the Dirac-Frenkel time-dependent variational principle with the multiple Davydov $D_{1}$ ansatz. A transformed RWA (TRWA) treatment and a standard perturbation (SP) are used to analytically calculate the emission spectrum. It is found that the variational approach and the TRWA treatment yield accurate emission spectra of the two distant qubits in certain strong coupling regimes while the SP breaks down. The emission spectrum is found to be asymmetric irrespective of the two-qubit distance and exhibits a single peak, doublet, and multipeaks depending on the two-qubit distance as well as the initial states. In sharply contrast with the single-qubit case, the excited-state populations of the two qubits can ultraslowly decay due to the subradiance even in the presence of a strong qubit-waveguide coupling, which in turn yields ultranarrow emission line. Our results provide insights into the emission spectral features of the two distant qubits in the strong light-matter coupling regime.Comment: 15 pages, 4 figure

Improved Best-of-Both-Worlds Guarantees for Multi-Armed Bandits: FTRL with General Regularizers and Multiple Optimal Arms

We study the problem of designing adaptive multi-armed bandit algorithms that perform optimally in both the stochastic setting and the adversarial setting simultaneously (often known as a best-of-both-world guarantee). A line of recent works shows that when configured and analyzed properly, the Follow-the-Regularized-Leader (FTRL) algorithm, originally designed for the adversarial setting, can in fact optimally adapt to the stochastic setting as well. Such results, however, critically rely on an assumption that there exists one unique optimal arm. Recently, Ito (2021) took the first step to remove such an undesirable uniqueness assumption for one particular FTRL algorithm with the $\frac{1}{2}$-Tsallis entropy regularizer. In this work, we significantly improve and generalize this result, showing that uniqueness is unnecessary for FTRL with a broad family of regularizers and a new learning rate schedule. For some regularizers, our regret bounds also improve upon prior results even when uniqueness holds. We further provide an application of our results to the decoupled exploration and exploitation problem, demonstrating that our techniques are broadly applicable.Comment: Update the camera-ready version for NeurIPS 202

Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by $\kappa>0$ whose solution converges to that of the MHD equations as $\kappa\to 0$. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.Comment: 60 page