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Le problème d’équivalence locale pour un système scalaire complet d’équations aux dérivées partielles d’ordre deux à n variables indépendantes

Camille Bièche (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans le présent article, nous établissons une caractérisation des systèmes scalaires d’équations aux dérivées partielles analytiques d’ordre deux à n variables indépendantes équivalents par un changement de coordonnées analytique au système u x α x β = 0 , 1 ≤ α , β ≤ n .

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation ∂ t u - ∂ ³ t x x u + 2 κ ∂ x u + ∂ x [ g ( u ) / 2 ] = γ ( 2 ∂ x u ∂ ² x x u + u ∂ ³ x x x u ) for the initial data u₀(x) in the Besov space B p , r s ( ℝ ) with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given C m -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids

Irena Pawłow, Wojciech M. Zajączkowski (2008)

Colloquium Mathematicae

The paper is concerned with the existence of measure-valued solutions to the Cahn-Hilliard system coupled with elasticity. The system under consideration is anisotropic and heterogeneous in the sense of admitting the elasticity and gradient energy tensors dependent on the order parameter. Such dependences introduce additional nonlinearities to the model for which the existence of weak solutions is not known so far.

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