2,305 research outputs found
Global Strong Solutions of the Boltzmann Equation without Angular Cut-off
We prove the existence and exponential decay of global in time strong
solutions to the Boltzmann equation without any angular cut-off, i.e., for
long-range interactions. We consider perturbations of the Maxwellian
equilibrium states and include the physical cross-sections arising from an
inverse-power intermolecular potential with , and more
generally, the full range of angular singularities . These
appear to be the first unique global solutions to this fundamentally important
model, which grants a basic example where a range of geometric fractional
derivatives occur in a physical model of the natural world. Our methods provide
a new understanding of the effects of grazing collisions in the Boltzmann
theory.Comment: This file has not changed, but this work has been combined with
(arXiv:1002.3639v1), simplified and extended into a new preprint, please see
the updated version: arXiv:1011.5441v
Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production
This article provides sharp constructive upper and lower bound estimates for
the non-linear Boltzmann collision operator with the full range of physical non
cut-off collision kernels ( and ) in the trilinear
energy . These new estimates prove that, for
a very general class of , the global diffusive behavior (on ) in the
energy space is that of the geometric fractional derivative semi-norm
identified in the linearized context in our earlier works [2009, 2010, 2010
arXiv:1011.5441v1]. We further prove new global entropy production estimates
with the same anisotropic semi-norm. This resolves the longstanding, widespread
heuristic conjecture about the sharp diffusive nature of the non cut-off
Boltzmann collision operator in the energy space .Comment: 29 pages, updated file based on referee report; Advances in
Mathematics (2011
A non-local inequality and global existence
In this article we prove a collection of new non-linear and non-local
integral inequalities. As an example for and we
obtain \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed}
dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. We
use these inequalities to deduce global existence of solutions to a non-local
heat equation with a quadratic non-linearity for large radial monotonic
positive initial conditions. Specifically, we improve \cite{ksLM} to include
all .Comment: 6 pages, to appear in Advances in Mathematic
Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials
In this work we prove global stability for the Boltzmann equation (1872) with
the physical collision kernels derived by Maxwell in 1866 for the full range of
inverse power intermolecular potentials, with . This
completes the work which we began in (arXiv:0912.0888v1). We more generally
cover collision kernels with parameters and satisfying
in arbitrary dimensions
with . Moreover, we prove rapid convergence as predicted by the
Boltzmann H-Theorem. When , we have exponential time decay
to the Maxwellian equilibrium states. When , our solutions
decay polynomially fast in time with any rate. These results are constructive.
Additionally, we prove sharp constructive upper and lower bounds for the
linearized collision operator in terms of a geometric fractional Sobolev norm;
we thus observe that a spectral gap exists only when , as
conjectured in Mouhot-Strain (2007).Comment: This file has not changed, but this work has been combined with
(arXiv:0912.0888v1), simplified and extended into a new preprint, please see
the updated version: arXiv:1011.5441v
Optimal time decay of the non cut-off Boltzmann equation in the whole space
In this paper we study the large-time behavior of perturbative classical
solutions to the hard and soft potential Boltzmann equation without the angular
cut-off assumption in the whole space \threed_x with \DgE. We use the
existence theory of global in time nearby Maxwellian solutions from
\cite{gsNonCutA,gsNonCut0}. It has been a longstanding open problem to
determine the large time decay rates for the soft potential Boltzmann equation
in the whole space, with or without the angular cut-off assumption
\cite{MR677262,MR2847536}. For perturbative initial data, we prove that
solutions converge to the global Maxwellian with the optimal large-time decay
rate of O(t^{-\frac{\Ndim}{2}+\frac{\Ndim}{2r}}) in the
L^2_\vel(L^r_x)-norm for any .Comment: 31 pages, final version to appear in KR
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
We study the Hilbert expansion for small Knudsen number for the
Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term
takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{\rho_{0}(t,x)}{(2\pi
\theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\theta_{0}(t,x)},\text{\
}\theta_{0}(t,x)=K\rho_{0}^{2/3}(t,x).t=0u_00\leq t\leq \varepsilon
^{-{1/2}\frac{2k-3}{2k-2}},\rho_{0}(t,x) u_{0}(t,x)\gamma=5/3$
Hilbert Expansion from the Boltzmann equation to relativistic Fluids
We study the local-in-time hydrodynamic limit of the relativistic Boltzmann
equation using a Hilbert expansion. More specifically, we prove the existence
of local solutions to the relativistic Boltzmann equation that are nearby the
local relativistic Maxwellian constructed from a class of solutions to the
relativistic Euler equations that includes a large subclass of near-constant,
non-vacuum fluid states. In particular, for small Knudsen number, these
solutions to the relativistic Boltzmann equation have dynamics that are
effectively captured by corresponding solutions to the relativistic Euler
equations.Comment: 50 page
Electroreduction and photometric detection of low-level uranium in aqueous Purex solutions. Consolidated Fuel-Reprocessing Program
During proper operation of the Purex process for the recovery of uranium and plutonium from spent reactor fuel, there are only trace levels of uranium in the aqueous waste. In the event of an upset in the extraction columns the aqueous waste stream would give the first indication of breakthrough. From the standpoint of process control it would be desirable to have an in-line, real-time sensor for uranium in the aqueous waste stream. It was toward this end that this investigation was undertaken. The measurement technique that seems to provide the most sensitive method without addition of reagents appears to be the electrochemical reduction of UO{sub 2}{sup 2+} to U(IV) followed by spectral measurement. The electrochemical reduction to U(IV) increases the sensitivity by a factor of three to five and shifts the measurement wavelength to a spectral area (647 nm and 1075 nm) unaffected by fission products. Using the proposed analysis sequence it is possible to determine uranium at a level of 0.2 g/L in the presence of relatively high spectral background. This report details the electrochemical reduction of U(VI) in nitric acid solutions (0.5 M to 2.0 M) with platinum-vitreous carbon electrodes and examines the spectral behavior of U(IV) as a function of nitric acid concentration
Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
In this paper it is shown that unique solutions to the relativistic Boltzmann
equation exist for all time and decay with any polynomial rate towards their
steady state relativistic Maxwellian provided that the initial data starts out
sufficiently close in . If the initial data are continuous then
so is the corresponding solution. We work in the case of a spatially periodic
box. Conditions on the collision kernel are generic in the sense of
(Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves
the open question of global existence for the soft potentials.Comment: 64 page
Experimental demonstration of a suspended diffractively coupled optical cavity
All-reflective optical systems are under consideration for future gravitational wave detector topologies. One approach in proposed designs is to use diffraction gratings as input couplers for Fabry–Perot cavities. We present an experimental demonstration of a fully suspended diffractively coupled cavity and investigate the use of conventional Pound–Drever–Hall length sensing and control techniques to maintain the required operating condition
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