Integrability and regularity of 3D Euler and equations for uniformly rotating fluids

AbstractWe consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ(t, x1, x2) + V(t, x1, x2, x3) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical 3). Here r is a remainder of order Ro12a where Roa = (H0U0(Щ0L20) is the anisotropic Rossby number (H0—height, L0—horizontal length scale, Щ0—angular velocity of background rotation, U0—horizontal velocity scale); Roa = (H0L0)Ro where H0L0 is the aspect ratio and Ro = U0(Щ0L0) is a Rossby number based on the horizontal length scale L0. The vector field V(t, x1, x2, x3) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Щ0t, τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e3 and velocity Ū3(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x1, x2, x3) and 2D turbulence for vertically averaged fields Ū(t, x1, x2) if the anisotropic Rossby number Roa is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T∗ of regular solutions to 3D Euler equations (T∗ → +∞, as Roa → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case

Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy

The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data

A unique classical solution of the Cauchy problem for the Navier-Stokes equations is considered when the initial velocity is spatially almost periodic. It is shown that the solution is always spatially almost periodic at any time provided that the solution exists. No restriction on the space dimension is imposed. This fact follows from continuous dependence of the solution with respect to initial data in uniform topology. Similar result is also established for Cauchy problem of the three-dimensional Navier-Stokes equations in a rotating frame

Hydrothermal Surface-Wave Instability and the Kuramoto-Sivashinsky Equation

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrothermal surface-wave instabilities may give rise to disturbances governed by the Kuramoto-Sivashinsky equation. A possible connection to hot-wire experiments is also discussed.Comment: 11 pages, RevTex, no figure

Mercury content in soils on the territory of Mezhdurechensk

The geochemical features of mercury content and distribution in the zone of coal producers have been studied (Mezhdurechensk town). Mercury content in soil (30 samples) was determined by atomic absorption method using mercury analyzer PA-915+ with pyrolytic device. Mercury content in soil samples changed from 0.12 to 0.17 mg/kg, the average value being 0.057 mg/kg. Within the town territory five zones with mercury elevated concentrations in soil were distinguished. 25-year observation period showed a 2.8 time decrease in average mercury content in soil. The major contribution to soil pollution in the urban territory was made by the two factors: local and regional. The mercury content in soil is affected by the emissions from boilers operating on coal as well as coal dust from the open pits near the town

Energy‐Dependent Boltzmann Equation in the Fast Domain

This work presents some aspects of the static energy‐dependent Boltzmann equation in plane geometry using a continuous‐energy formulation. In a first part, solutions are found for a class of synthetic separable (but nondegenerate) energy‐transfer kernels. Such kernels are representative, for instance, of neutron inelastic slowing down. In a second part, the same problem is considered with the addition of a projection kernel (typical of neutron fission); it is shown that the solutions split into space‐energy separable components and nonseparable ``slowing‐down transients.''Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69987/2/JMAPAQ-11-1-174-1.pd

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

STROBE: Stake-based Threshold Random Beacons

We revisit decentralized random beacons with a focus on practical distributed applications. Decentralized random beacons (Beaver and So, Eurocrypt 1993) provide the functionality for $n$ parties to generate an unpredictable sequence of bits in a way that cannot be biased, which is useful for any decentralized protocol requiring trusted randomness. Existing beacon constructions are highly inefficient in practical settings where protocol parties need to rejoin after crashes or disconnections, and more significantly where smart contracts may rely on arbitrary index points in high-volume streams. For this, we introduce a new notion of history-generating decentralized random beacons (HGDRBs). Roughly, the history-generation property of HGDRBs allows for previous beacon outputs to be efficiently generated knowing only the current value and the public key. At application layers, history-generation supports registering a sparser set of on-chain values if desired, so that apps like lotteries can utilize on-chain values without incurring high-frequency costs, enjoying all the benefits of DRBs implemented off-chain or with decoupled, special-purpose chains. Unlike rollups, HG is tailored specifically to recovering and verifying pseudorandom bit sequences and thus enjoys unique optimizations investigated in this work. We introduce STROBE: an efficient HGDRB construction which generalizes the original squaring-based RSA approach of Beaver and So. STROBE enjoys several useful properties that make it suited for practical applications that use beacons: - history-generating: it can regenerate and verify high-throughput beacon streams, supporting sparse (thus cost-effective) ledger entries; - concisely self-verifying: NIZK-free, with state and validation employing a single ring element; - eco-friendly: stake-based rather than work based; - unbounded: refresh-free, addressing limitations of Beaver and So; - delay-free: results are immediately available

Differitial diagnosis of sinoatrial blockade in rhythmocardiography

Researches had studied in arrhythmology during 5-8 years for evaluation of high-resolution rhythmocardiography (RCG) possibilities for sinoatrial blockade differential diagnosis. 362 patients (pts) (270 with dysfunction of sinus node-DSN- and 92 with sick sinus syndrome -SSS). The differential signs were defined between at functional sinus breaches and morphological SSS. They have different autonomic background, DSN has more safe wave HRV structure, and at the SSS was define syndrome autonomic cardioneuropathy with stabilization of the HRV and life-dangerous arrhythmias, for example –atrial fibrillation. RCG shows hemodynamic meaning of every arrhythmic episode, that is very important in treatment.Проводились исследования в аритмологии для оценки возможностей метода ритмокардиографии высокого разрешения (РКГ) для анализа вариабельности сердечного ритма в (ВСР) дифференциальной диагностике синоатриальной блокады. В течение 5-8 лет проводились РКГ-обследования 362 пациентов с дисфункцией синоатриального узла (ДФСУ) на фоне ишемической болезни сердца (ИБС). Из их числа выделены 92 случая с синдромом слабости синусового узла (СССУ). Найдены дифференцированные признаки между функциональными нарушениями (ДФСУ) и синоатриальной блокадой органического генеза, их автономного фона и гемодинамического значения аритмических эпизодов

Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model

The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-B\'enard convection model. The mathematical analysis the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations