×

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
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] A. Alvino - P.L. Lions - G. Trombetti , On optimization problems with prescribed rearrangement. Nonlinear Analysis Theory , Methods and Applications . Vol. 13 N^\circ 2 ( 1989 ), 185 - 220 . MR 979040 | Zbl 0678.49003 · Zbl 0678.49003
[2] C. Bandle , Isoperimetric Inequalities and Applications . Pitman Advances Publishing Program , Boston - London - Melboume 1980 . MR 572958 | Zbl 0436.35063 · Zbl 0436.35063
[3] K.M. Chong - N.M. Rice , Equimeasurable rearrangement of functions . Queen’s University ( 1971 ). MR 372140 | Zbl 0275.46024 · Zbl 0275.46024
[4] P. Faurre , Analyse Numérique Notes d’optimisation . Ecole polytechnique ( 1984 ). MR 971252 · Zbl 0912.65048
[5] I.V. Girsanov , Lectures on mathematical theory of extremum problems . Springer , Berlin , 1972 . MR 464021 | Zbl 0234.49016 · Zbl 0234.49016
[6] B. Kawhol , On rearrangements, symmetrization and maximum principles . Lectures Notes in Math . 1150 , Springer-Verlag , Berlin , 1985 .
[7] S. Kurcyusz , On the existence and nonexistence of Lagrange multipliers in Banach spaces . J. Optim. Theory Appl. 20 ( 1976 ), 81 - 110 . MR 424249 | Zbl 0309.49010 · Zbl 0309.49010
[8] J. Mossino , Inégalités isoperimétriques et application en Physique. Collection Travaux en cours , Hermann , Paris , 1984 . MR 733257 | Zbl 0537.35002 · Zbl 0537.35002
[9] J. Mossino - R. Temam , Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics . Duke Math. J. 48 ( 1981 ), 475 - 495 . Article | MR 630581 | Zbl 0476.35031 · Zbl 0476.35031
[10] J. Mossino - J.M. Rakotoson , Isoperimetric inequalities in Parabolic equations . Ann. Scuola Norm. Sup. Pisa Cl. Sci . ( 4 ) 13 ( 1986 ), 51 - 73 . Numdam | MR 863635 | Zbl 0652.35053 · Zbl 0652.35053
[11] J.M. Rakotoson , Réarrangement relatif et équations aux dérivées partielles. Cours de 3ème cycle . Université de Poitiers Année 1990 - 1991 (Graduate classes).
[12] J.M. Rakotoson - R. Temam , A co-area formula with applications to monotone rearrangement and to regularity . Arch. Rational Mech. Anal. 109 ( 1990 ), 213 - 238 . MR 1025171 | Zbl 0735.49039 · Zbl 0735.49039
[13] D. Serre , Sur le principe variationnel des équations de la mécanique des fluides parfaits . Modélisation mathématique et analyse numérique 27 ( 1993 ), 207 - 226 . Numdam | MR 1246997 | Zbl 0788.76065 · Zbl 0788.76065
[14] J. Zowe - S. Kurcyusz , Regularity and stability for the mathematical programming problem in Banach space . Applied Math. Optim. 5 ( 1979 ), 49 - 62 . MR 526427 | Zbl 0401.90104 · Zbl 0401.90104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.