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Transport equations due to the non-Lipschitzian vector fields and fluid mechanics. (Equations de transport relatives à des champs de vecteurs non- lipschitziens et mécanique des fluides.) (French) Zbl 0821.76012
The authors study the properties of transport equations due to logarithmic Lipschitzian vector fields, i.e. equations of the type \(\partial_ tf + \text{div} (fv) = g\), \(f |_{t=0} = f_ 0\). They prove the existence of a unique solution in certain function spaces and show that such vector fields possess a flow whose Hölder regularity is exponentially decreasing.

76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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[1] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales de l’Ecole Normale Supérieure, 14, 1981, pages 209-246.
[2] J.-Y. Chemin, Une facette mathématique de la mécanique des fluides I, Prépublication n? 1055 de l’Ecole Polytechnique, 1993.
[3] R. DiPerna & P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98, 1989, pages 511-549. · Zbl 0696.34049 · doi:10.1007/BF01393835
[4] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970). · Zbl 0207.13501
[5] H. Triebel, Interpolation theory, function spaces, differential operators, North Holland (1978). · Zbl 0387.46032
[6] W. Wolibner, Un théorème d’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Mathematische Zeitschrift, 37, 1933, pages 698-726. · JFM 59.1447.02 · doi:10.1007/BF01474610
[7] M. Yamasaki, A quasi-homogeneous version of paradifferential operators, I. Bounded on spaces of Besov type, Journal of the Faculty of Science of the University of Kyoto, 33, pages 131-174.
[8] V. Yudovitch, Non stationnary flow of an ideal and incompressible liquid, Zh. Vych. Math, 3, 1963, pages 1032-1066.
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