Existence of weak solutions for unsteady motions of generalized Newtonian fluids

Diening, Lars¹  ; Růžička, Michael¹  ; Wolf, Jörg²

¹ Universität Freiburg, Mathematisches Institut, Eckerstr, 1, 79104 Freiburg, Germany
² Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, Postfach 4120, 39106 Magdeburg, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 1-46

Résumé

We prove the existence of weak solutions $𝐮:Q_T\to {ℝ}^n$ of the equations of unsteady motion of an incompressible fluid with shear-dependent viscosity in a cylinder $Q_T=\Omega \times (0,T)$, where $\Omega \subset {ℝ}^n$ denotes a bounded domain. Under the assumption that the extra stress tensor $𝐒$ possesses a $q$-structure with $q>\frac{2n}{n+2}$, we are able to construct a weak solution $𝐮\in L^q(0,T;W_0^{1,q}\left(\Omega \right))\cap C_w([0,T];L^2\left(\Omega \right))$ with $div𝐮=0$. Our approach is based on the Lipschitz truncation method, which is new in this context.

Détail

Classification : 76D03, 35D05, 35D46, 34A34