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On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

We consider the 3D quantum BBGKY hierarchy which corresponds to the N -particle Schrödinger equation. We assume the pair interaction is N 3 β – 1 V ( B β ) . For the interaction parameter β ∈ ( 0 , 2 / 3 ) , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the N → ∞ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for β ∈ ( 0 , 3 / 5 ) in [28]. This allows, in the case β ∈ ( 0 , 3 / 5 ) , for the application of the Klainerman–Machedon uniqueness theorem...

On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem

Hugo Beirão da Veiga (2009)

Journal of the European Mathematical Society

We establish regularity results up to the boundary for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case p ≥ 2 . Actually, we are interested in proving regularity results in L q ( Ω ) spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced...

On the Lawrence–Doniach model of superconductivity: magnetic fields parallel to the axes

Stan Alama, Lia Bronsard, Etienne Sandier (2012)

Journal of the European Mathematical Society

We consider periodic minimizers of the Lawrence–Doniach functional, which models highly anisotropic superconductors with layered structure, in the simultaneous limit as the layer thickness tends to zero and the Ginzburg–Landau parameter tends to infinity. In particular, we consider the properties of minimizers when the system is subjected to an external magnetic field applied either tangentially or normally to the superconducting planes. For normally applied fields, our results show that the resulting...

On the linear force-free fields in bounded and unbounded three-dimensional domains

Tahar-Zamène Boulmezaoud, Yvon Maday, Tahar Amari (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Linear Force-free (or Beltrami) fields are three-components divergence-free fields solutions of the equation curlB = αB, where α is a real number. Such fields appear in many branches of physics like astrophysics, fluid mechanics, electromagnetics and plasma physics. In this paper, we deal with some related boundary value problems in multiply-connected bounded domains, in half-cylindrical domains and in exterior domains.

On the linear problem arising from motion of a fluid around a moving rigid body

Šárka Matušů-Nečasová, Jörg Wolf (2015)

Mathematica Bohemica

We study a linear system of equations arising from fluid motion around a moving rigid body, where rotation is included. Originally, the coordinate system is attached to the fluid, which means that the domain is changing with respect to time. To get a problem in the fixed domain, the problem is rewritten in the coordinate system attached to the body. The aim of the present paper is the proof of the existence of a strong solution in a weighted Lebesgue space. In particular, we prove the existence...

On the local strong solutions for a system describing the flow of a viscoelastic fluid

Ondřej Kreml, Milan Pokorný (2009)

Banach Center Publications

We consider a model for the viscoelastic fluid which has recently been studied in [4] and [1]. We show the local-in-time existence of a strong solution to the corresponding system of partial differential equations under less regularity assumptions on the initial data than in the above mentioned papers. The main difference in our approach is the use of the L p theory for the Stokes system.

On the localization of the vortices

Carlo Marchioro (1998)

Bollettino dell'Unione Matematica Italiana

Studiamo l'evoluzione temporale di un fluido bidimensionale incomprimibile non viscoso quando la vorticità iniziale è concentrata in N regioni di diametro ϵ e mostriamo che la vorticità evoluta temporalmente è anche lei concentrata in N piccole regioni di diametro d , d ≤ const ⁢ ϵ α per qualunque α < 1 / 3 . Noi chiamiamo questa proprietà "localizzazione". Come conseguenza abbiamo una connessione rigorosa tra il modello dei vortici puntiformi e l'Equazione di Eulero.

On the long time behavior of KdV type equations

Nikolay Tzvetkov (2003/2004)

Séminaire Bourbaki

In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg- de Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.

On the long-time behaviour of compressible fluid flows subjected to highly oscillating external forces

Sergiu Aizicovici, Eduard Feireisl (2003)

Czechoslovak Mathematical Journal

We show that the global-in-time solutions to the compressible Navier-Stokes equations driven by highly oscillating external forces stabilize to globally defined (on the whole real line) solutions of the same system with the driving force given by the integral mean of oscillations. Several stability results will be obtained.

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