Well-posedness of a multiscale model for concentrated suspensions

In a previous work [math.AP/0305408] three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and as a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker-Planck type equation with nonhomogeneous, nonlocal and possibly degenerate, coefficients. We show the existence and the uniqueness of the global-in-time weak solution to such a system.Comment: 1 figur

Enhanced binding revisited for a spinless particle in non-relativistic QED

We consider a spinless particle coupled to a quantized Bose field and show that such a system has a ground state for two classes of short-range potentials which are alone too weak to have a zero-energy resonance

Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree–Fock equations: A sufficient condition

AbstractThe multiconfiguration time-dependent Hartree–Fock (MCTDHF for short) system is an approximation of the linear many-particle Schrödinger equation with a binary interaction potential by nonlinear “one-particle” equations. MCTDHF methods are widely used for numerical calculations of the dynamics of few-electron systems in quantum physics and quantum chemistry, but the time-dependent case still poses serious open problems for the analysis, e.g. in the sense that global-in-time existence of solutions is not proved yet. In this letter we present the first result ever where global existence is proved under a condition on the initial datum that it has to be somewhat close to the “ground state”

Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations

In this paper we motivate, formulate and analyze the Multi-Configuration Time-Dependent Hartree-Fock (MCTDHF) equations for molecular systems under Coulomb interaction. They consist in approximating the N-particle Schrodinger wavefunction by a (time-dependent) linear combination of (time-dependent) Slater determinants. The equations of motion express as a system of ordinary differential equations for the expansion coefficients coupled to nonlinear Schrodinger-type equations for mono-electronic wavefunctions. The invertibility of the one-body density matrix (full-rank hypothesis) plays a crucial role in the analysis. Under the full-rank assumption a fiber bundle structure shows up and produces unitary equivalence between convenient representations of the equations. We discuss and establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as long as the density matrix is not singular. A sufficient condition in terms of the energy of the initial data ensuring the global-in-time invertibility is provided (first result in this direction). Regularizing the density matrix breaks down energy conservation, however a global well-posedness for this system in L^2 is obtained with Strichartz estimates. Eventually solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible.Comment: 48 pages, 1 figur

Self-energy of one electron in non-relativistic QED

Abstract. We investigate the self-energy of one electron coupled to a quantized radiation field by extending the ideas developed in [H]. We fix an arbitrary cut-off parameter Λ and recover the α 2-term of the selfenergy, where α is the coupling parameter representing the fine structure constant. Thereby we develop a method which allows to expand the selfenergy up to any power of α. This implies that perturbation theory is correct if Λ is fix. As an immediate consequence we obtain enhanced binding for electrons. 1

Properties of periodic Dirac--Fock functional and minimizers

Existence of minimizers for the Dirac--Fock model in crystals was recently proved by Paturel and S\'er\'e and the authors \cite{crystals} by a retraction technique due to S\'er\'e \cite{Ser09}. In this paper, inspired by Ghimenti and Lewin's result \cite{ghimenti2009properties} for the periodic Hartree--Fock model, we prove that the Fermi level of any periodic Dirac--Fock minimizer is either empty or totally filled when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. Here $c$ is the speed of light, $\alpha$ is the fine structure constant, and $C_{\rm cri}$ is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for $C_{\rm cri}$. Our result implies that any minimizer of the periodic Dirac--Fock model is a projector when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. In particular, the non-relativistic regime (i.e., $c\gg1$) and the weak coupling regime (i.e., $0<\alpha\ll1$) are covered. The proof is based on a delicate study of a second-order expansion of the periodic Dirac--Fock functional composed with the retraction used in \cite{crystals}

Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-newtonian flows

add proof of uniqueness of steady statesThe mathematical properties of a nonlinear parabolic equation arising in the modelling of non-newtonian flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact a linear advection equation). Depending on the initial data, at least two situations can be encountered: the equation may have a unique solution in a convenient class, or it may have infinitely many solutions