Long time behaviour of viscous scalar conservation laws

This paper is concerned with the stability of stationary solutions of the conservation law $\partial_t u + \mathrm{div}_y A(y,u) -\Delta_y u=0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviours are studied here: the case when the equation is set on $\R$, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of viscous stationary shocks - also called standing shocks - which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable in $L^1$, provided the initial disturbance satisfies some appropriate boundedness conditions. We also extend this result to arbitrary initial data, but with some restrictions on the flux $A$. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniform $L^\infty$ bounds on the solution of the conservation law, and on sub- and super-solution techniques.Comment: 36 page

Periodically-driven quantum systems: Effective Hamiltonians and engineered gauge fields

Driving a quantum system periodically in time can profoundly alter its long-time dynamics and trigger topological order. Such schemes are particularly promising for generating non-trivial energy bands and gauge structures in quantum-matter systems. Here, we develop a general formalism that captures the essential features ruling the dynamics: the effective Hamiltonian, but also the effects related to the initial phase of the modulation and the micro-motion. This framework allows for the identification of driving schemes, based on general N-step modulations, which lead to configurations relevant for quantum simulation. In particular, we explore methods to generate synthetic spin-orbit couplings and magnetic fields in cold-atom setups.Comment: 25 pages, 6 figures, includes Appendices (A-K). An erroneous factor of two has been corrected in the last term of Eq. C10 (Appendix C); this typo had no impact on the rest of the articl

Relation between energy shifts and relaxation rates for a small system coupled to a reservoir

For a small system the coupling to a reservoir causes energy shifts as well as transitions between the system's energy levels. We show for a general stationary situation that the energy shifts can essentially be reduced to the relaxation rates. The effects of reservoir fluctuations and self reaction are treated separately. We apply the results to a two-level atom coupled to a reservoir which may be the vacuum of a radiation field.Comment: 6 pages, Latex, to appear in Phys. Lett.

Well-posedness of the Stokes-Coriolis system in the half-space over a rough surface

This paper is devoted to the well-posedness of the stationary $3$d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of G\'erard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space $H^{1/2}_{uloc}$.Comment: 64 page

Operator Ordering in Quantum Radiative Processes

In this work we reexamine quantum electrodynamics of atomic eletrons in the Coulomb gauge in the dipole approximation and calculate the shift of atomic energy levels in the context of Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC) formalism by considering the variation rates of physical observables. We then analyze the physical interpretation of the ordering of operators in the dipole approximation interaction Hamiltonian in terms of field fluctuations and self-reaction of atomic eletrons, discussing the arbitrariness in the statistical functions in second order bound-state perturbation theory.Comment: Latex file, 12 pages, no figures, includes PACS numbers and minor changes in the text with the addition of a new sectio