On the decay of infinite energy solutions to the Navier-Stokes equations in the plane

Infinite energy solutions to the Navier-Stokes equations in R² may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon (2002) [2] as well as solutions constructed from a “radial energy decomposition”. Our proof uses the Fourier Splitting technique of M.E. Schonbek.     

Radiation damping as an initial value problem

Using a simple, exactly soluble model for the interaction of one particle and a scalar field Φ, we discuss the problem of radiation reaction in terms of the initial value solution. We show that if the Cauchy data of the field fall off at spatial infinity in such a way that the field has finite energy, the particle motion is damped for t → ∞. Further, we point out that no solutions with finite field energy exist for the boundary conditions Φ^out = 0 and Φⁱⁿ + Φ^out = 0. For Φⁱⁿ = 0, nontrivial solutions exist only if it is assumed that the system has been open in the past of some initial hypersurface.     

The Quantum and Klein-Gordon Oscillators in a Non-commutative Complex Space and the Thermodynamic Functions

In this work we study the quantum and Klein-Gordon oscillators in a non-commutative complex space. We show that a particle described by such oscillators behaves similarly as an electron with spin in a commutative space in an external uniform magnetic field. Therefore the wave-function $\psi (z,\bar{z} )$ takes values in C ⁴, spin up, spin down, particle, antiparticle, a result which is obtained by the Dirac theory. We obtain the energy levels by exact solutions. We also derive the thermodynamic functions associated to the partition function, and show that the non-commutativity effects are manifested in energy at the high temperature limit.     

Mass, energy, and the electron

The two-component solutions of the Dirac equation currently in use are not separately a particle equation or an antiparticle equation. We present a unitary transformation that uncouples the four-component, force-free Dirac equation to yield a two-component spinor equation for the force-free motion of a relativistic particle and a corresponding two-component, time-reversed equation for an antiparticle. The particle-antiparticle nature of the two equations is established by applying to the solutions of these two-component equations criteria analogous to those applied for establishing the four-component particle and antiparticle solutions of the four-component Dirac equation. Wave function solutions of our two-component particle equation describe both a right and a left circularly polarized particle. Interesting characteristics of our solutions include spatial distributions that are confined in extent along directions perpendicular to the motion, without the artifice of wave packets, and an intrinsic chirality (handedness) that replaces the usual definition of chirality for particles without mass. Our solutions demonstrate that both the rest mass and the relativistic increase in mass with velocity of the force-free electron are due to an increase in the rate of Zitterbewegung with velocity. We extend this result to a bound electron, in which case the loss of energy due to binding is shown to decrease the rate of Zitterbewegung.     

On Isotropic Distributional Solutions to the Boltzmann Equation for Bose-Einstein Particles

The paper considers the spatially homogeneous Boltzmann equation for Bose-Einstein particles (BBE). In order to include the hard sphere model, the equation is studied in a weak form and its solutions (including initial data) are set in the class of isotropic positive Borel measures and therefore called isotropic distributional solutions. Stability of distributional solutions is established in the weak topology, global existence of distributional solutions that conserve the mass and energy is proved by weak convergence of approximate L ¹-solutions, and moment production estimates for the distributional solutions are also obtained. As an application of the weak form of the BBE equation, it is shown that a Bose-Einstein distribution plus a Dirac dt-function is an equilibrium solution to the BBE equation in the weak form if and only if it satisfies a low temperature condition and an exact ratio of the Bose-Einstein condensation.     

A theorem on the existence of dyon solutions

We prove the existence of finite energy dyon solutions to Yang-Mills-Higgs equations satisfying the Julia-Zee ansatz, and the generalization to SU(N) gauge groups. This rigorously establishes the existence of a model for the particles having electric and magnetic charge conjectured by Schwinger. We also prove that the solutions are real analytic on (0, ∞) and C^∞ at r = 0. To establish our result we prove a new abstract theorem that allows one to study singular constrained minimization problems without the introduction of Lagrange multipliers.     

Stability of the Boltzmann equation with potential forces on torus

In this paper, we are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in L¹-norm for any initial data with the finite total mass, mechanical energy and entropy.     

Approximate Relativistic Bound State Solutions of the Tietz–Hua Rotating Oscillator for Any κ-State

Approximate analytical solutions of the Dirac equation with Tietz–Hua (TH) potential are obtained for arbitrary spin–orbit quantum number κ using the Pekeris approximation scheme to deal with the spin–orbit coupling terms κ(κ ± 1)r ^?2. In the presence of exact spin and pseudo-spin symmetric limitation, the bound state energy eigenvalues and associated two-component wave functions of the Dirac particle moving in the field of attractive and repulsive TH potential are obtained using the parametric generalization of the Nikiforov–Uvarov method. The cases of the Morse potential, the generalized Morse potential and non-relativistic limits are studied.     

Orbital Stability of Dirac Solitons

We prove H ¹ orbital stability of Dirac solitons in the integrable massive Thirring model by working with an additional conserved quantity which complements Hamiltonian, momentum and charge functionals of the general nonlinear Dirac equations. We also derive a global bound on the H ¹ norm of the L ²-small solutions of the massive Thirring model.     

Quasi-Linear Dynamics in Nonlinear Schrödinger Equation with Periodic Boundary Conditions

It is shown that a large subset of initial data with finite energy (L ² norm) evolves nearly linearly in nonlinear Schrödinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such as solitons, semiclassical or weakly linear solutions.     

On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations

In this paper, we study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see Definition 2.2). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (Comm Pure Appl Math 51:229–240, 1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions.     

The Boltzmann Equation for Bose–Einstein Particles: Regularity and Condensation

We study regularity and finite time condensation of distributional solutions of the space-homogeneous and velocity-isotropic Boltzmann equation for Bose–Einstein particles for the hard sphere model. Global in time existence of distributional solutions had been proven before. Here we prove that the equation is locally and can be globally (in time) well-posed for the class of distributional solutions having finite moment of the negative order \(-1/2\) , and solutions in this class with regular initial data are mild solutions in their regularity time-intervals. By observing a necessary condition on the initial data for the absence of condensation at some finite time, we also propose a sufficient condition on the initial data for the occurrence of condensation at all large time, and then using a positivity of a partial collision integral we prove further that the critical time of condensation can be strictly positive.     

Blowup of Smooth Solutions for Relativistic Euler Equations

We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial ``generalized' momentum is sufficiently large.     

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns L ¹-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L ¹-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|γ|. For the usual L ¹-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L ¹-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates. E.A. Carlen work partially supported by US National Science Foundation grant DMS 06-00037. M.C. Carvalho work partially supported by POCI/MAT/61931/2004. X. Lu work partially supported by NSF of China grant 10571101.     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

On the spectrum of particles created in a Robertson-Walker universe

Created particle spectra are calculated in Robertson-Walker universes with scale factora∝t ^α(0 <α≦ 1) anda∝e ^t, and discussed with a special emphasis on their dependence upon the initial and final times at which a WKB-like positive frequency condition should be imposed. It is shown that the obtained spectra are very sensitive to these times if the WKB approximation for the field equation is not valid in their neighborhood. It is also shown that the total number density of created particles remains finite if the final time is set to be finite.     

Dissipative or conservative cosmology with dark energy?

All evolutional paths for all admissible initial conditions of FRW cosmological models with dissipative dust fluid (described by dark matter, baryonic matter and dark energy) are analyzed using dynamical system approach. With that approach, one is able to see how generic the class of solutions leading to the desired property—acceleration—is. The theory of dynamical systems also offers a possibility of investigating all possible solutions and their stability with tools of Newtonian mechanics of a particle moving in a one-dimensional potential which is parameterized by the cosmological scale factor. We demonstrate that flat cosmology with bulk viscosity can be treated as a conservative system with a potential function of the Chaplygin gas type. We characterize the class of dark energy models that admit late time de Sitter attractor solution in terms of the potential function of corresponding conservative system. We argue that inclusion of dissipation effects makes the model more realistic because of its structural stability. We also confront viscous models with SNIa observations. The best fitted models are obtained by minimizing the χ² function which is illustrated by residuals and χ² levels in the space of model independent parameters. The general conclusion is that SNIa data supports the viscous model without the cosmological constant. The obtained values of χ² statistic are comparable for both the viscous model and ΛCDM model. The Bayesian information criteria are used to compare the models with different power-law parameterization of viscous effects. Our result of this analysis shows that SNIa data supports viscous cosmology more than the ΛCDM model if the coefficient in viscosity parameterization is fixed. The Bayes factor is also used to obtain the posterior probability of the model.     

Relativistic effects in one-dimensional hydrogen atom

The dependence of stationary levels of a Dirac particle in the regularized Coulomb potential V _??(z) = ?q/(|z| + ??) on the cutoff parameter ?? is studied. It is shown that, in 1 + 1 D, the energy spectrum of a Dirac particle in such a potential reveals some specific features which nonanalytically depend on the coupling constant q and are essentially relativistic in nature. These properties turn out to be most important for ?? ? 1, explicitly demonstrating the existence of a physically reasonable energy spectrum for an arbitrarily small ?? > 0 and, at the same time, the absence of regular limit ?? ?? 0 (hence, the absence of a well-defined spectral problem for the Dirac equation without regularization for arbitrary q in 1 + 1 D).     

Solutions of Dirac Equation in the Presence of Modified Tietz and Modified Poschl-Teller Potentials Plus a Coulomb-Like Tensor Interaction Using SUSYQM

The solutions of the Dirac equation with Modified Tietz and Modified Poschl-Teller scaler and vector potentials including the tensor interaction term for arbitrary spin-orbit quantum number κ are presented. We obtained the energy eigenvalues and the corresponding wave functions using the supersymmetry method. To show the accuracy of our results, we calculate the energy eigenvalues numerical for different values of n and κ. It is shown that these results are in good agreement with those found in the literature.