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We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain $Ω$ , which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $Γ_{-}$ and $Γ_{+}$ (lower and upper parts of $\partial Ω$ ), the Dirichlet boundary conditions on $Γ_{in}$ (the inflow) and $Γ_{0}$ (boundary of the profile) and an artificial “do nothing”-type boundary condition...

The Maxwell equation in a periodic medium : homogenization of the energy density

P. A. Markowich, F. Poupaud (1996)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The mean-field limit for the dynamics of large particle systems

François Golse (2003)

Journées équations aux dérivées partielles

This short course explains how the usual mean-field evolution PDEs in Statistical Physics - such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations - are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.

The microlocal Landau-Zener formula

Yves Colin de Verdière, Maurice Lombardi, Joël Pollet (1999)

Annales de l'I.H.P. Physique théorique

The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section

Dietmar Kröner, Philippe G. LeFloch, Mai-Duc Thanh (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl.74 (1995) 483–548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class...

The mixed regularity of electronic wave functions multiplied by explicit correlation factors

Harry Yserentant (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics 2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...

The mixed regularity of electronic wave functions multiplied by explicit correlation factors***

Harry Yserentant (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...

The modulational regime of three-dimensional water waves and the Davey-Stewartson system

Walter Craig, Ulrich Schanz, Catherine Sulem (1997)

Annales de l'I.H.P. Analyse non linéaire

The motion of a fluid in an open channel

Simina Bodea (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a free boundary value problem for a viscous, incompressible fluid contained in an uncovered three-dimensional rectangular channel, with gravity and surface tension, governed by the Navier-Stokes equations. We obtain existence results for the linear and nonlinear time-dependent problem. We analyse the qualitative behavior of the flow using tools of bifurcation theory. The main result is a Hopf bifurcation theorem with $ℤ_{k}$ -symmetry.

The motion of the free surface of a liquid

Hans Lindblad (2000/2001)

Séminaire Équations aux dérivées partielles

The Mourre estimate for regular dispersive systems

C. Gérard (1991)

Annales de l'I.H.P. Physique théorique

The Navier-Stokes equation with inhomogeneous boundary conditions based on vorticity

Jiří Neustupa, Patrick Penel (2008)

Banach Center Publications

We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.

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