TY - JOUR
AB - The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.
AU - László Erdös
AU - Solovej, Jan P
ID - 2741
IS - 4
JF - Annales Henri Poincare
TI - Uniform Lieb-Thirring inequality for the three-dimensional Pauli operator with a strong non-homogeneous magnetic field
VL - 5
ER -
TY - JOUR
AB - We consider a system of N weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle density matrix of this system and the solution of the nonlinear Hartree equation is of order N−1 for any time t⩽T.
AU - Elgart, Alexander
AU - László Erdös
AU - Schlein, Benjamin
AU - Yau, Horng-Tzer
ID - 2742
IS - 10
JF - Journal de Mathématiques Pures et Appliquées
TI - Nonlinear Hartree equation as the mean field limit of weakly coupled fermions
VL - 83
ER -
TY - JOUR
AB - Transition to turbulence in pipe flow is one of the most fundamental and longest- standing problems in fluid dynamics. Stability theory suggests that the flow remains laminar for all flow rates, but in practice pipe flow becomes turbulent even at moderate speeds. This transition drastically affects the transport efficiency of mass, momentum, and heat. On the basis of the recent discovery of unstable traveling waves in computational studies of the Navier-Stokes equations and ideas from dynamical systems theory, a model for the transition process has been suggested. We report experimental observation of these traveling waves in pipe flow, confirming the proposed transition scenario and suggesting that the dynamics associated with these unstable states may indeed capture the nature of fluid turbulence.
AU - Björn Hof
AU - van Doorne, Casimir W
AU - Westerweel, Jerry
AU - Nieuwstadt, Frans T
AU - Faisst, Holger
AU - Eckhardt, Bruno
AU - Wedin, Håkan
AU - Kersweli, Richard R
AU - Waleffe, Fabian
ID - 2786
IS - 5690
JF - Science
TI - Experimental observation of nonlinear traveling waves in turbulent pipe flow
VL - 305
ER -
TY - JOUR
AB - The results of experimental and numerical investigations of the onset of oscillatory convection in a sidewall heated rectangular cavity of molten gallium are reported. Detailed comparisons are made between experimental observations and calculations from numerical simulations of a three-dimensional Boussinesq model. The onset of time-dependence takes place through supercritical Hopf bifurcations and the loci of critical points in the (Gr, Pr)-plane are qualitatively similar with excellent agreement between the frequencies of the oscillatory motion. This provides a severe test of the control of the experiment since the mode of oscillation is extremely sensitive to imperfections. Detailed numerical investigations reveal that there are a pair of Hopf bifurcations which exist on two asymmetric states which themselves arise at a subcritical pitchfork from the symmetric state. There is no evidence for this in the experiment and this qualitative difference is attributed to non-Boussinesq perturbations which increase with Gr. However, the antisymmetric spatial structure of the oscillatory state is robust and is present in both the experiment and the numerical model. Moreover, the detailed analysis of the numerical results reveals the origins of the oscillatory instability.
AU - Björn Hof
AU - Juel, Anne
AU - Zhao, Li
AU - Henry, Daniel
AU - Ben Hadid, Hamda
AU - Mullin, Tom P
ID - 2787
JF - Journal of Fluid Mechanics
TI - On the onset of oscillatory convection in molten gallium
VL - 515
ER -
TY - JOUR
AB - We consider the evolution of a connected set on the plane carried by a space periodic incompressible stochastic flow. While for almost every realization of the stochastic flow at time t most of the particles are at a distance of order equation image away from the origin, there is a measure zero set of points that escape to infinity at the linear rate. We study the set of points visited by the original set by time t and show that such a set, when scaled down by the factor of t, has a limiting nonrandom shape.
AU - Dolgopyat, Dmitry
AU - Kaloshin, Vadim
AU - Koralov, Leonid
ID - 8517
IS - 9
JF - Communications on Pure and Applied Mathematics
KW - Applied Mathematics
KW - General Mathematics
SN - 0010-3640
TI - A limit shape theorem for periodic stochastic dispersion
VL - 57
ER -