Asymptotic Analysis

In this paper, we study the convergence of strong solutions of a MHD system. The proofs, in booth cases of ${\mathbb{T}}^{3}$ or ${\mathbb{R}}^{3}$ , are based on spectral properties of the penalized operator and the energy method. Moreover, in the case of the whole space, we prove a refined convergence result for initial data 2D + 3D.

We derive the complete asymptotic series, as t→+∞, for a general solution h(t) of the nonlinear differential equation h³(h″+h′)=1. The equation originates from a physical model related to the Hall effect.

In this paper we study the behaviour of the Landau–Lifschitz equation in a thin layer. As the thickness of the domain and the exchange coefficient of the equation simultaneously tend to zero we perform an asymptotic expansion to precise the solution for well prepared initial condition.

The classical problem of homogenization deals with elliptic operators in periodically oscillating media of small period ε>0 and the asymptotic behavior of solution u^ε of boundary value problems associated with such operators. In a previous work [Conca et al., SIAM J. Math. Anal. 33 (2002), 1166–1198], the authors introduced what is called Bloch approximation which provided energy norm approximation for the solution in $\mathbb{R}^{N}$ . This paper continues with the above development and proposes a modified Bloch approximation. This function takes into account boundary effects. Its connection with the first order classical correctors is also established with the corresponding error estimate. All the proofs are worked out entirely in the Fourier–Bloch space.