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Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

Mireille Bossy, Nicolas Champagnat, Sylvain Maire, Denis Talay (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of ℝ d . This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator...

Problème de Stokes et système de Navier-Stokes incompressible à densité variable dans le demi-espace

Raphaël Danchin, Piotr Bogusław Mucha (2008/2009)

Séminaire Équations aux dérivées partielles

On s’intéresse à la résolution du système de Navier-Stokes incompressible à densité variable dans le demi-espace ℝ + n : = ℝ n - 1 × ] 0 , ∞ [ en dimension n ≥ 3 . On considère des données initiales à régularité critique. On établit que si la densité initiale est proche d’une constante strictement positive dans L ∞ ∩ W ˙ 1 , n et si la vitesse initiale est petite par rapport à la viscosité dans l’espace de Besov homogène B ˙ n , 1 0 alors le système de Navier-Stokes admet une unique solution globale. La démonstration repose sur de nouvelles estimations...

Profile decomposition for solutions of the Navier-Stokes equations

Isabelle Gallagher (2001)

Bulletin de la Société Mathématique de France

We consider sequences of solutions of the Navier-Stokes equations in  ℝ 3 , associated with sequences of initial data bounded in  H ˙ 1 / 2 . We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in  H ˙ 1 / 2 , up to a remainder term small in  L 3 ; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If  𝒜 is an “admissible” space (in particular ...

Proof of the Treves theorem on the KdV hierarchy

Leonid Dickey (2005)

Annales de l’institut Fourier

A new, shorter, proof of the Treves theorem on an algebraic criterion for the first integrals of the KdV hierarchy is given, along with an addition to the theorem.

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