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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