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- CONTENTS 1987 - VOLUME 305 - SECTION I - N° 11
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- .......... Page(s) .......... 449
- Fernandez and Hamilton have conjectured that if L is a regular curve in the sense of Ahlfors, U a simply connected domain of C, and f a conformal mapping from the unit disk onto U, then is bounded by a constant depending only on L. We prove that this conjecture is false for general regular sets but that it is true for curves in the case .
- An analysis of solutions with shock arising in elasticity and hydrodynamics models is given in this Note. These models have naturally a non-conservative form, as in the Hooke law in Eulerian coordinates. Up to now, any attempt to give them a conservative form has led to nonsense. The use of the multiplication of distributions allows to get conditions of the Rankine Hugoniot type, which rule the shock waves. We remark, by using this analysis, that the specific volume, the velocity and the pressure are moving in phase across a shock wave in hydrodynamics. By identifying the speed of the elastic and hydrodynamic shock waves, we get that the velocity and the strain also move in phase. These properties are needed to use performing numerical methods; they confirm some choices of numerical techniques found in elastoplastic codes for industrial applications.
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- We study some maximal functions on Lie groups of polynomial growth, and the atomic decomposition for Hardy spaces associated to heat semi-groups.
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- Given a germ, at , of multiform function F with real positive exponents, we prove first that each connected component of the hypersurfaces of F in is obtained from a ringed neighbourhood of with cw-complexes of index greater than or equal to n-1. When F has "an isolated singularity" a counter-example shows that certain expected results cannot be obtained.
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- We prove that the solution of a non linear Cauchy problem of order 2, where data have conormal singularities near a point, is smooth off the union of null bicharacteristic curves issued from that point.
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- We prove, for small initial data in , the existence and uniqueness of a global solution of a Dirac equation with cubic nonlinearity and zero mass.
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- We present a bootstrap argument that allows us to show that C2 solutions of Grad generalized differential equations in a bounded domain are , provided a lower bound on the modulus of the gradient can be obtained.
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- We obtain sufficient conditions of geometric mixing for law stationary processes in satisfying an equation: where is a polynomial application. To do this, we first prove a continuity theorem for the image of a measure by a polynomial application.
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- We use the Hermite's corrector formula for solving the systems of ordinary differential equations, in particular the Kepler's equations. By an analog method, we have an algorithm of fixed points.
- .......... Page(s) .......... 489
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- In this Note viscoelastic fluids which obey an Oldroyd type constitutive law with retardation time are considered. For the mixed Cauchy problem with homogeneous Dirichlet conditions, the local existence of smooth solutions is shown. Then a global existence result for solutions with small data is proved.
- We consider the Couette-Taylor problem at the neighbourhood of degenerated Hopf bifurcation, for which it is necessary to compute the seventh order terms in the normal form of the system on the center manifold. We describe a method which allows to compute these terms using symbolic calculus. So, we show the existence of new type of quasi-periodic flow, unfortunately unstable for the specified values of the parameters.
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- Our aim in this Note is to present some results concerning the interaction of small and large eddies in turbulent flows. We show that the amplitude of small structures decays exponentially to a small value and we infer from this a simplified interaction law of small and large eddies. Beside their intrinsic interest for the understanding of the physics of turbulence, these results lead to new numerical schemes which will be studied in a separate work.
- COMPTES RENDUS DE L'ACADEMIE DES SCIENCES
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