FLUIDES DE SECOND ET TROISIEME GRADE EN DIMENSION TROIS : SOLUTION GLOBALE ET REGULARITE

Abstract : This thesis focuses on the time-dependent problems of second and third grade fluids, in three dimensions, as regards global existence in time of the weak solution and its regularity. Also we consider a particular case of the stationary problem of second grade fluids, in three dimensions. We study these problems in a bounded domain of IR3, at first simply connected, subsequently multiply-connected, a case not considered in previous works. In the first part, we show that the decomposition method with special basis, introduced by D. Cioranescu and E. H. Ouazar, allows us to prove global existence in time of the weak solution for second grade fluids in the general case ( a1 + a2 different from 0), in three dimensions, with small data. Contrary to the simpler case, where a1 + a2 = 0, recently studied by D. Cioranescu and V. Girault, the exponential decay with respect to time of the H1 norm of velocity is not obtained for all data. This fact, which led some authors to assert, in contradiction to our work, that the method of decomposition does not apply to the case where a1+a 2 different from 0, complicates substantially the proof of the existence of the solution. The regularity results, which lead, in particular, to a classical solution, are less straightforward than in the case where a1 + a2 = 0 because of a transport equation, which is much more complex. The second part is devoted to the stationary problem of second grade, in the case where a1 + a2 = 0, in three dimensions. In relation to the problem in two dimensions, studied by E. H. Ouazar, the H3 norm of the velocity, in three dimensions, is not bounded for all data. However, by a special method, using together a H1 bound of the velocity, a "pseudo continuous dependence" with respect to the data (effective for a small H3 norm of the velocity) and a polynomial inequality (verified by the H3 norm of the velocity), we show existence, uniqueness, continuous dependence with respect to the data and regularity of the solution, with small data. Finally, the third part deals with the problem of third grade fluids, in three dimensions, but without assuming a condition, which, in the C. Amrouche and D. Cioranescu work, gave a H1 bound of the velocity for all data. By a method similar to that of the first part, given that the difficulties are the same, we obtain, with a few more technical complications, the same types of results as for the second grade fluids. iii
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Jean-Marie Bernard. FLUIDES DE SECOND ET TROISIEME GRADE EN DIMENSION TROIS : SOLUTION GLOBALE ET REGULARITE. Equations aux dérivées partielles [math.AP]. Université Paris VI, 1998. Français. ⟨tel-01361460⟩

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