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Existence and uniqueness for fluids of second grade. (English) Zbl 0577.76012
Nonlinear partial differential equations and their applications, Coll. de France Semin., Paris 1982-83, Vol. VI, Res. Notes Math. 109, 178-197 (1984).
[For the entire collection see Zbl 0543.00005.]
Consider the following initial boundary value problem (IVP) for the motion of a second grade fluid $\partial u/\partial t-\nu \Delta u- \alpha (\partial /\partial t)\Delta u+\text{curl}(u-\alpha \Delta u)\wedge u=f-\nabla p;\quad \text{div } u=0$ where $$f$$ is given and $$p=\alpha (u\cdot \Delta u+1/4| \nabla u|^ 2)-1/2| u|^ 2-\tilde p;\quad u=0$$ on $$\partial \Omega;\quad u(x,0)=u_ 0(x)$$ and $$\Omega$$ is a domain in $${\mathbb R}^ n.$$
Let $$V$$ be the closure in $$[H^ 1(\Omega)]^ n$$ of the divergence free vectors in $$[{\mathcal D}(\Omega)]^ n$$ and let $$W$$ be the divergence free vectors in $$[H^ 3(\Omega)]^ n$$ vanishing on $$\partial \Omega$$. Let $$T>0$$ be given. Then, for $$\Omega \subset \mathbb R^ 2$$, $$f$$ given in $$L^ 2(0,T;V)$$ and $$u_ 0$$ in $$V$$, the authors prove the existence and uniqueness of solutions in $$L^{\infty}(0,T;W)$$ for the IVP. For $$\Omega \subset \mathbb R^ 3$$ a bounded domain, $$f$$ in $$L^ 2(0,T;V)$$ and $$u_ 0$$ in $$W$$, existence and uniqueness of solutions in $$L^{\infty}(0,T^*;W)$$ for the IVP were proved for $$T^*<T$$ sufficiently small.
Reviewer: Dang Dinh Ang

##### MSC:
 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics