Decomposition of the pressure of a fluid in a thin domain and applications: Asymptotic behavior of fluids and elasticity problems.

Jeudi 27 septembre 2018, 11:30 à 12:30

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Manuel Luna-Laynez

(Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla)

This is a joint work with J. Casado-Díaz and F.J. Suárez-Grau.

The $L^q$ norm of the pressure of a fluid in a thin domain of small thickness $\varepsilon$ is usually estimated by the $W^{-1,q}$ norm of its gradient, multiplied by a positive constant which is of order $1/\varepsilon$. In this talk we provide an improvement of this estimate by showing that the pressure is the sum of two terms, one of them of order $1$ and the other one of order $1/\varepsilon$ with respect to norm of the gradient, but with the advantage that the term of order $1/\varepsilon$ does not only belong to $L^q$ but to $W^{1,q}$. This result extends to the linear elasticity framework, which allows to obtain better estimates for the Korn's constant and gives fine decompositions of the elastic deformations of thin domains. We show how these decomposition results can be applied to study the asymptotic behavior of some problems in Fluid Mechanics and in Elasticity.