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2D-1D dimensional reduction in a toy model for magnetoelastic interactions

Mouhcine Tilioua (2011)

Applications of Mathematics

The paper deals with the dimensional reduction from 2D to 1D in magnetoelastic interactions. We adopt a simplified, but nontrivial model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem by using the so-called energy method.

30 Years of Calderón’s Problem

Gunther Uhlmann (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

In this article we survey some of the most important developments since the 1980 paper of A.P. Calderón in which he proposed the problem of determining the conductivity of a medium by making voltage and current measurements at the boundary.

A 2D model for hydrodynamics and biology coupling applied to algae growth simulations

Olivier Bernard, Anne-Céline Boulanger, Marie-Odile Bristeau, Jacques Sainte-Marie (2013)

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

Cultivating oleaginous microalgae in specific culturing devices such as raceways is seen as a future way to produce biofuel. The complexity of this process coupling non linear biological activity to hydrodynamics makes the optimization problem very delicate. The large amount of parameters to be taken into account paves the way for a useful mathematical modeling. Due to the heterogeneity of raceways along the depth dimension regarding temperature, light intensity or nutrients availability, we adopt...

A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Huanyuan Li (2021)

Applications of Mathematics

This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ρ and velocity field u satisfy ∥ ∇ ρ ∥ L ∞ ( 0 , T ; W 1 , q ) + ∥ u ∥ L s ( 0 , T ; L ω r ) < ∞ for some q > 3 and any ( r , s ) satisfying 2 / s + 3 / r ≤ 1 , 3 < r ≤ ∞ , then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [ 0 , T ] . Here L ω r denotes the weak L r space.

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