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The so-called lake equations arise as the shallow-water limit of the rigid-lid equations—three-dimensional Euler equations with a rigid-lid upper boundary condition—in a horizontally periodic basin with bottom topography. We prove an a priori estimate in the Sobolev space H m for m≥ 3 which shows that a solution to the rigid-lid equations can be approximated by a solution of the lake equations for an interval of time which can be estimated in terms of the initial deviation from a columnar configuration and the magnitude of the initial data in H m , the gradient of the bottom topography in H m+1 , and the aspect ratio of the basin. In particular, any solution to the lake equations remains close to some solution of the rigid-lid equations for an interval of time that can be made arbitrarily large by choosing the aspect ratio of the basin small. Received 10 October 1996 and accepted 15 May 1997  相似文献   
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The consecutive reactions of (CH3)2Si(OC2H5)2 and CH3Si(OC2H5)3 with methoxide ions were investigated in methanol solutions. The reverse transesterification reactions with ethoxide ions could be neglected in both cases since the concentration of ethoxide in methanol solution was assumed to be low due to the fast equilibrium reaction C2H5O? + CH3OH ? C2H5OH + CH3O?. The progress of the reactions was followed by monitoring the formation of ethanol with a Fourier-transform infrared spectrometer. All rate constants were determined at 295 K. The reactions between the dialkoxydimethylsilanes and methoxide ions were assumed to consist of two consecutive steps that can be represented by the net reaction; (CH3)2Si(OC2H5)2 + 2CH3O? → (CH3)2Si(OCH3)2 + 2C2H5O?. The two consecutive rate constants were established as 1.93 ± 0.12M?1s?1 and 1.00 ± 0.12M?1s?1, respectively. The consecutive rate constants for the reactions between the trialkoxymethylsilanes and methoxide ions can be written according to the total reaction; CH3Si(OC2H5)3 + 3CH3O? → CH3Si(OCH3)3 + 3C2H5O?. The three rate constants corresponding to each consecutive step were established as 1.12 ± 0.09 M?1s?1, 0.82 ± 0.10 M?1s?1, and 0.51 ± 0.06 M?1s?1, respectively. © 1995 John Wiley & Sons, Inc.  相似文献   
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We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates. The authors are members of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.  相似文献   
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Ohne Zusammenfassung  相似文献   
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We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ2, and form the Voronoi tessellation of ℝ2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p>1/2 then the union of the black cells contains an infinite component with probability 1, while if p<1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in ℤ2. The result corresponding to Harris' Theorem for bond percolation in ℤ2 is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for ℤ2; we hope they will be applicable in other contexts as well. Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900 Research partially undertaken during a visit to the Forschungsinstitut für Mathematik, ETH Zürich, Switzerland  相似文献   
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