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Low Mach number limit for viscous compressible flows

Raphaël Danchin (2005)

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

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes equations...

Low Mach number limit for viscous compressible flows

Raphaël Danchin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes...

Lyapunov functions and L p -estimates for a class of reaction-diffusion systems

Dirk Horstmann (2001)

Colloquium Mathematicae

We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, ε c ₜ = k c Δ c - f ( c ) c + g ( a , c ) , x ∈ Ω, t > 0, for Ω ⊂ ℝ N , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform L p -estimates.

Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint

Ayman Kachmar (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is devoted to an analysis of vortex-nucleation for a Ginzburg-Landau functional with discontinuous constraint. This functional has been proposed as a model for vortex-pinning, and usually accounts for the energy resulting from the interface of two superconductors. The critical applied magnetic field for vortex nucleation is estimated in the London singular limit, and as a by-product, results concerning vortex-pinning and boundary conditions on the interface are obtained.

Morse index and blow-up points of solutions of some nonlinear problems

Khalil El Mehdi, Filomena Pacella (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we consider the following problem - △ ⁢ u = N ⁢ N - 2 ⁢ u p ϵ - λ ⁢ u in  ⁢ Ω u > 0 in  ⁢ Ω u = 0 on  ⁢ ∂ ⁡ Ω . where Ω is a bounded smooth starshaped domain in R N , N ≥ 3 , p ϵ = N + 2 N - 2 - ϵ , ϵ > 0 , and λ ≥ 0 . We prove that if u ϵ is a solution of Morse index m > 0 than u ϵ cannot have more than m maximum points in Ω for ϵ sufficiently small. Moreover if Ω is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for ϵ sufficiently small.

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