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Sur un problème d’optimisation lié aux équations de Navier-Stokes. (French) Zbl 0797.49006
According to D. Serre’s conjecture, the flux current \(\phi_ 0\) associated to a bidimensional Navier-Stokes equation in a bounded domain \(\Omega\) satisfies the following minimization problem: \(J(\phi_ 0)= \text{Inf}\{J(\phi),\phi\in {\mathcal K}(h)\}\), where \(J(\phi)= {1\over 2} \int_ \Omega |\nabla \phi|^ 2- \int_ \Omega f\phi\), \(f\in L^ \infty(\Omega)\), \(h\in L^ 1(\Omega)\) and \({\mathcal K}(h)\) is the set \(\{v\in H_ 0'(\Omega)\), \(\int_ \Omega \psi(v)\leq \int_ \Omega \psi(h)\), \(\forall \psi: \mathbb{R}\to\mathbb{R}\) convex and Lipschitzian}. We show in this paper that the solution \(\phi_ 0\) is in \(W^{2,p}(\Omega)\) for all \(p< +\infty\). Furthermore, we give the Euler equation associated to \(\phi_ 0\). The method uses an abstract theorem amering the existence of Lagrange multipliers. The application of this abstract theorem to the above optimization problem needs the notion of relative rearrangement.
Reviewer: J.M.Rakotoson

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
35Q30 Navier-Stokes equations
49J35 Existence of solutions for minimax problems
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