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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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References:
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