Uniqueness results for monopolar fluids (Q1570382)

In a bounded sufficiently smooth domain \(\Omega\in\mathbb{R}^n\), \(n\geq 2\), the author considers the non-stationary motion of a power-law fluid with the constitutive equation \(\tau(\nu)= 2\mu|e(\nu)|^{p- 2}e\), where \(\tau\) is shear stress, and \(e\) is deformation rate tensor. If \(f\) is the external force and \(\nu\) is the fluid velocity, the main result states that for \(f\in L^{p/(p- 1)}\), \(\nu_0\in H\) and \(p> n\), \(n\geq 2\), the corresponding initial-boundary value problem has a unique solution \(\nu\in L^p\). The existence result is well known [see, e.g., \textit{J. L. Lions}, Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; Gauthier-Villars. XX (1969; Zbl 0189.40603)], and here the author proves only the uniqueness. The proof is based on Sobolev imbedding theorems and on the Korn inequality.