By S. Friedlander, D. Serre
ISBN-10: 0444515569
ISBN-13: 9780444515568
The guide of Mathematical Fluid Dynamics is a compendium of essays that offers a survey of the foremost themes within the topic. every one article lines advancements, surveys the result of the earlier decade, discusses the present country of data and offers significant destiny instructions and open difficulties. vast bibliographic fabric is supplied. The e-book is meant to be worthy either to specialists within the box and to mathematicians and different scientists who desire to find out about or start learn in mathematical fluid dynamics. The guide illuminates a thrilling topic that includes rigorous mathematical thought utilized to an enormous actual challenge, particularly the movement of fluids.
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Extra resources for Handbook of mathematical fluid dynamics. / Vol. 3
Sample text
32) for a system of N particles interacting via a smooth, short range potential of range ε. Note that this is exactly the scaling we have postulated in the previous section, on the basis of heuristic arguments. 32) is usually called a low density limit or, in macroscopic variables, the Boltzmann–Grad limit. 32). From particles to fluids 27 Instead of considering the usual BBGKY hierarchy however, we derive another more convenient equation for a sequence of quantities slightly differing from the marginal distributions, but asymptotically equivalent to them.
We shall remark later this important aspects. 59). Suppose that initially the sequence of distribution functions {f0,j }∞ j =1 factorizes, namely j f0,j (x1 , v1 , . . , xk , vk ) = f0 (xk , vk ). 59) can be produced by putting j fj (x1 , v1 , . . 63) is the Boltzmann collision operator. 59) is called the Boltzmann hierarchy. 61) (propagation of chaos) states that if initially position and momentum of a given particle are distributed independently of the positions and momenta of all the others, such property is maintained during the time evolution.
72) and fjε (Xj , Vj , t) the solution of this hierarchy. e. where f (t) is the unique (mild) solution of the Boltzmann equation. R EMARK 1. We have formulated the result for cutoff cross section. The general case can be handled with a minor effort assuming a Gaussian decay on the velocities for the initial datum. R EMARK 2. The time for which the result holds is proportional to λ which is the mean free path, actually it is a fraction of the mean free time. This is of course very unsatisfactory.
Handbook of mathematical fluid dynamics. / Vol. 3 by S. Friedlander, D. Serre
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