Finite Kinetic Energy Research Articles

A note on the validity of Bogoliubov correction to mean-field dynamics

We study the norm approximation to the Schrödinger dynamics of N bosons in R3 with an interaction potential of the form N3β−1w(Nβ(x−y)). Assuming that in the initial state the particles outside of the condensate form a quasi-free state with finite kinetic energy, we show that in the large N limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation for all 0≤β<1/2. The range of β is expected to be optimal for this large class of initial states.

Global Existence to the Vlasov–Poisson System and Propagation of Moments Without Assumption of Finite Kinetic Energy

We consider classical as well as weak solutions to the three dimensional Vlasov–Poisson system. Without assuming finiteness of kinetic energy, we prove global existence of classical solutions by assuming the initial datum is smooth enough and has a compact velocity-spatial support, which will be specified in Theorem 1.1. We also establish some propagation results for low moments of weak solutions.

Inertial Motions of a Rigid Body with a Cavity Filled with a Viscous Liquid

We study inertial motions of the coupled system, \({\mathscr{S}}\), constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (a la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of \({\mathscr{S}}\) coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of \({\mathscr{S}}\) with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation.

Current densities in density-functional theory

It is well known that any given density rho(x)can be realized by a determinantal wave function for N particles. The question addressed here is whether any given density rho(x) and current density j(x) can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field v(x) =j(x)/rho(x) is curl free, we provide a solution for all N, and we provide an explicit upper bound for the energy. If the velocity field is not curl free, there is a finite energy solution for all N\geq 4, but we do not provide an explicit energy bound in this case. For N=2 we provide an example of a non curl free velocity field for which there is a solution, and an example for which there is no solution. The case $N=3 with a non curl free velocity field is left open.

Ultrahigh energy collision with neither black hole nor naked singularity

We explore the collision between two concentric spherical thin shells. The inner shell is charged, whereas the outer one is either neutral or charged. In the situation we consider, the charge of the inner shell is larger than its gravitational mass, and the inside of it is empty and regular. Hence the domain just outside it is described by the overcharged Reissner-Nordstrom geometry whereas the inside of it is Minkowski. First, the inner shell starts to shrink form infinity with finite kinetic energy, and then the outer shell starts to shrink from infinity with vanishing kinetic energy. The inner shell bounces on the potential wall and collides with the ingoing outer shell. The energy of collision between these shells at "their center of mass frame" does not exceed the total energy of the system. By contrast, by virtue of the very large gamma factor of the relative velocity of the shells, the energy of collision between two of the constituent particles of these shells at their center of mass frame can be much larger than the Planck scale. This result suggests that the black hole or naked singularity is not necessary for ultra-high energy collision of particles.

Self-Similar Solutions of the Compressible Flow in One-Space Dimension

For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.

Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles

We proved existence, regularity and uniqueness of steady subsonic potential flows in n-dimensional (n ≥ 3) infinite nozzles with largely-open convergent and divergent parts when the total mass flux is less than a certain value. Such nozzles consist of two cones with arbitrary open angles and an arbitrary smooth bounded tubular part. The existence of a weak solution is proved by applying the direct method of calculus of variation to a carefully chosen functional defined on a Hilbert space based upon Hardy inequality. Holder gradient regularity of weak solution is shown by using Moser iteration to quasilinear elliptic equations in divergence form. Also, the obtained solution is unique in the class of functions with finite kinetic energy by modulo a constant.

Triggering turbulence efficiently in plane Couette flow

AbstractWe use a variational formulation incorporating the full Navier–Stokes equations to identify initial perturbations with finite kinetic energy ${E}_{0} $ which generate the largest gain in perturbation kinetic energy at some time $T$ later for plane Couette flow. Using the flow geometry originally used by Butler &amp; Farrell (Phys. Fluids A, vol. 4, 1992, pp. 1637–1650) to identify the linear transient optimal perturbations for ${E}_{0} \ensuremath{\rightarrow} 0$ and incorporating $T$ as part of the optimization procedure, we show how the addition of nonlinearity smoothly changes the result as ${E}_{0} $ increases from zero until a small but finite ${E}_{c} $ is reached. At this point, the variational algorithm is able to identify an initial condition of completely different form which triggers turbulence – called the minimal seed for turbulence. If instead $T$ is fixed at some asymptotically large value, as suggested by Pringle, Willis &amp; Kerswell (J. Fluid Mech., vol. 703, 2012, pp. 415–443), a fundamentally different ‘final’ optimal perturbation emerges from our algorithm above some threshold initial energy ${E}_{f} \in (0, {E}_{c} )$ which shows signs of localization. This nonlinear optimal perturbation clearly approaches the structure of the minimal seed as ${E}_{0} \ensuremath{\rightarrow} { E}_{c}^{\ensuremath{-} } $, although for ${E}_{0} \lt {E}_{c} $, its maximum gain over all time intervals is always less than the equivalent maximum gain for the ‘quasi-linear optimal perturbation’, i.e. the finite-amplitude manifestation of the underlying linear optimal perturbation. We also consider a wider flow geometry recently studied by Monokrousos et al. (Phys. Rev. Lett., vol. 106, 2011, 134502) and present evidence that the critical energy for transition ${E}_{c} $ they found by using total dissipation over a time interval as the optimizing functional is recovered using energy gain at a fixed target time as the optimizing functional, with the same associated minimal seed emerging. This emphasizes that the precise form of the functional does not appear to be important for identifying ${E}_{c} $ provided it takes heightened values for turbulent flows, as postulated by Pringle, Willis &amp; Kerswell (J. Fluid Mech., vol. 703, 2012, pp. 415–443). All our results highlight the irrelevance of the linear energy gain optimal perturbation for predicting or describing the lowest-energy flow structure which triggers turbulence.

On the cometary flow equations with force fields

The Cauchy problem of a nonlinear kinetic equation modeling the time evolution of a cometary flow interacting with a force field is discussed, two kinds of existence results for weak solutions are established for initial data having finite mass and finite kinetic energy. The first one concerns a given force field which is assumed to be divergence free with respect to the velocity variable, it is shown that there exists a nonnegative weak solution to the Cauchy problem when the initial datum and the force field have reasonable integrability. As a special case, we also consider a Lorentz field and give another type of existence result. The second one deals with self-consistent electrostatic field, we show that when the initial datum has an L2 integrability the system has a global nonnegative solution which extends a previous result obtained by one of the authors.

Temperature-dependent compressibility in graphene and two-dimensional systems

We calculate the finite temperature compressibility for two-dimensional semiconductor systems, monolayer graphene, and bilayer graphene within the Hartree-Fock approximation. We find that the calculated temperature dependent compressibility including exchange energy is non-monotonic. In 2D systems at low temperatures the inverse compressibility decreases first with increasing temperature, but after reaching a minimum it increases as temperature is raised further. At high enough temperatures the negative compressibility of low density systems induced by the exchange energy becomes positive due to the dominance of the finite temperature kinetic energy. The inverse compressibility in monolayer graphene is always positive and its temperature dependence appears to be reverse of the 2D semiconductor systems, i.e., it increases first with temperature and then decreases at high temperatures. The inverse compressibility of bilayer graphene shows the same non-monotonic behavior as ordinary 2D systems, but at high temperatures it approaches a constant which is smaller than the value of the non-interacting bilayer graphene. We find the leading order temperature correction to the compressibility within Hartree-Fock approximation to be $T^2 \ln T$ at low temperatures for all three systems.

Existence and uniqueness of time-periodic solutions with finite kinetic energy for the Navier–Stokes equations in

Our aim is to prove existence and uniqueness of time-periodic strong solutions with finite kinetic energy for the Navier–Stokes equations in . For this, appropriate conditions are imposed on the external force, together with a smallness condition involving the viscosity of the fluid and the period of motion. We extend the method we have recently used to construct steady states with finite kinetic energy to the time-periodic case. First, existence and uniqueness of strong solutions with finite kinetic energy are established for a linearized version of the problem, using the Galerkin method and the Fourier transform in the space variables. Then, a strong solution with finite kinetic energy for the nonlinear problem is obtained by means of the contraction mapping principle. We also show that such a solution satisfies the energy equality and is unique within a class of weak solutions.

Optical probe of strong correlations in LaNiO3 thin films

The optical properties of LaNiO3 thin films are investigated over a wide energy and temperature range. Thin films of varying thickness were grown by pulsed laser deposition on LaAlO3 and SrTiO3 substrates. The optical conductivity data of the films reveal a number of interband transitions above 1 eV, which are in good agreement with band structure calculations. No well defined Drude peak is observed; however, in stark contrast with local-density approximation theory predicting a finite density of states at the Fermi energy. This experimental finding of a vanishing Drude spectral weight, compared to a finite electron kinetic energy obtained from band structure calculations, highlights the importance of strong electronic correlations in LaNiO3.

Role of long waves in the stability of the plane wake

This work is directed toward investigating the fate of three-dimensional long perturbation waves in a plane incompressible wake. The analysis is posed as an initial-value problem in space. More specifically, input is made at an initial location in the downstream direction and then tracing the resulting behavior further downstream subject to the restriction of finite kinetic energy. This presentation follows the outline given by Criminale and Drazin [W. O. Criminale and P. G. Drazin, Stud. Appl. Math. 83, 123 (1990)] that describes the system in terms of perturbation vorticity and velocity. The analysis is based on large scale waves and expansions using multiscales and multitimes for the partial differential equations. The multiscaling is based on an approach where the small parameter is linked to the perturbation property independently from the flow control parameter. Solutions of the perturbative equations are determined numerically after the introduction of a regular perturbation scheme analytically deduced up to the second order. Numerically, the complete linear system is also integrated. Since the results relevant to the complete problem are in very good agreement with the results of the first-order analysis, the numerical solution at the second order was deemed not necessary. The use for an arbitrary initial-value problem will be shown to contain a wealth of information for the different transient behaviors associated to the symmetry, angle of obliquity, and spatial decay of the long waves. The amplification factor of transversal perturbations never presents the trend--a growth followed by a long damping--usually seen in waves with wave number of order one or less. Asymptotical instability is always observed.

Steady solutions with finite kinetic energy for a perturbed Navier–Stokes system in [formula omitted

Consider a Navier–Stokes liquid filling the three-dimensional space exterior to a moving rigid body and subject to an external force. Using a coordinates system attached to the body, the equations of the fluid can be written in a time-independent domain, which results in a perturbed Navier–Stokes system where the extra terms depend on the velocity of the rigid body. In this paper, we consider the related whole space problem and construct a strong solution with finite kinetic energy for the corresponding steady-state equations. For this, appropriate conditions on the external force have to be imposed (for instance, that it is a function with compact support and null average) together with a smallness condition involving the viscosity of the fluid. First, a linearized version of the problem is analysed by means of the Fourier transform, and then a strong solution to the full nonlinear problem is obtained by a fixed point procedure. We also show that such a solution satisfies the energy equation and is unique within a certain class.

Chebychev optimized approximate deconvolution models of turbulence

If the Navier–Stokes equations are averaged with a local, spacial convolution type filter, ϕ ¯ = g δ ∗ ϕ , the resulting system is not closed due to the filtered nonlinear term uu ¯ . An approximate deconvolution operator D is a bounded linear operator which satisfies u = D ( u ¯ ) + O ( δ α ) , where δ is the filter width and α ⩾ 2 . Using a deconvolution operator as an approximate filter inverse, yields the closure uu ¯ = D ( u ¯ ) D ( u ¯ ) ¯ + O ( δ α ) . The residual stress of this model (and related models) depends directly on the deconvolution error, u - D ( u ¯ ) . This report derives deconvolution operators yielding an effective turbulence model, which minimize the deconvolution error for velocity fields with finite kinetic energy. We also give a convergence theory of deconvolution as δ → 0 , an ergodic theorem as the deconvolution order N → ∞ , and estimate the increase in accuracy obtained by parameter optimization. The report concludes with numerical illustrations.

Compactness properties for trace-class operators and applications to quantum mechanics

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics.

Mild solutions for the one-dimensional Nordström–Vlasov system

The Nordström–Vlasov system describes the evolution of a population of self-gravitating collisionless particles. We study the existence and uniqueness of mild solutions for the Cauchy problem in one dimension. This approach does not require any derivative for the initial particle density. For any initial particle density uniformly bounded with respect to the space variable by some function with finite kinetic energy and any initial smooth data for the field equation we construct a global solution, preserving the total energy. Moreover the solution propagates with finite speed. The propagation speed coincides with the light speed.

Revisiting the first-principles approach to the granular gas steady state

We extend a Fokker-Planck formalism, previously used to describe the behavior of a cooling granular gas, with a Hertzian contact potential and viscoelastic radial friction, giving a velocity dependent coefficient of restitution. In the present work, we study the more general case of a steady-state with finite kinetic energy, far from equilibrium, due to the coupling to an external energy-feeding mechanism. Also from first-principles, we extend the validity of the former results.

Electromechanical influence of crack velocity at bifurcation for poled ferroelectric materials

The piezoelastodynamic field equations are solved to determine the crack velocity at bifurcation for poled ferroelectric materials where the applied electrical field and mechanical stress can be varied. The underlying physical mechanism, however, may not correspond to that assumed in the analytical model. Bifurcation has been related to the occurrence of a pair of maximum circumferential stress oriented symmetrically about the moving crack path. The velocity at which this behavior prevails has been referred to as the limiting crack speed. Unlike the classical approach, bifurcation will be identified with finite distances ahead of a moving crack. Nucleation of microcracks can thus be modelled in a single formulation. This can be accomplished by using the energy density function where fracture initiation is identified with dominance of dilatation in relation to distortion. Poled ferroelectric materials are selected for this study because the microstructure effects for this class of materials can be readily reflected by the elastic, piezoelectic and dielectric permittivity constants at the macroscopic scale. Existing test data could also shed light on the trend of the analytical predictions. Numerical results are thus computed for PZT-4 and compared with those for PZT-6B in an effort to show whether the branching behavior would be affected by the difference in the material microstructures. A range of crack bifurcation speed v b is found for different r/ a and E/ σ ratios. Here, r and a stand for the radial distance and half crack length, respectively, while E and σ for the electric field and mechanical stress. For PZT-6B with v b in the range 100–1700 m/s, the bifurcation angles varied from ±6° to ±39°. This corresponds to E/ σ of −0.072 to 0.024 V m/N. At the same distance r/ a=0.1, PZT-4 gives v b values of 1100–2100 m/s; bifurcation angles of ±15° to ±49°; and E/ σ of −0.056 to 0.059 V m/N. In general, the bifurcation angles ± θ 0 are found to decrease with decreasing crack velocity as the distance r/ a is increased. Relatively speaking, the speed v b and angles ± θ 0 for PZT-4 are much greater than those for PZT-6B. This may be attributed to the high electromechanical coupling effect of PZT-4. Using v b 0 as a base reference, an equality relation v b −< v b 0< v b + can be established. The superscripts −, 0 and + refer, respectively, to negative, zero and positive electric field. This is reminiscent of the enhancement and retardation of crack growth behavior due to change in poling direction. Bifurcation characteristics are found to be somewhat erratic when r/ a approaches the range 10 −2–10 −1 where the kinetic energy densities would fluctuate and then rise as the distance from the moving crack is increased. This is an artifact introduced by the far away condition of non-vanishing particle velocity. A finite kinetic energy density prevails at infinity unless it is made to vanish in the boundary value problem. Future works are recommended to further clarify the physical mechanism(s) associated with bifurcation by means of analysis and experiment. Damage at the microscopic level needs to be addressed since it has been known to affect the macrocrack speeds and bifurcation characteristics.

An existence theorem in the large for zero-pressure gas dynamics

We obtain existence of a weak solution of the equations of zero-pressure gas dynamics, in any number of space dimensions plus time, given initial data of finite mass and kinetic energy. Our solution satisfies the "sticky particle" or adhesion dynamics conditions. The argument depends on a definition of a weak solution based on the introduction of Lagrangian coordinates and in particular on an application of the entropy condition proposed by C. Dafermos for hyperbolic systems of conservation laws.