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Example text
Then we can prove Theorem. The integral operator K is in JB(L2) nJB(L~,L~+l)',B ~ JB(L2, La) and in C(L 2). o. It is also in Proof. For the proof see Ref. 4. We next consider the spectrum of L. Since to prove the theorems in this section and the following ones several standard theorems on the perturbation of linear operators are needed, for the sake of the reader we state them here and refer to the book of Kato I8 for the proofs. We denote by K( ... ) the theorem (... ) in Kato's book. In the statements below (where only the numbers of the equations and some symbols have been modified with respect to Ref.
6) Eq. 5) is, in particular, valid for steady problems. Although the idea of a scattering kernel had appeared before, it is only at the end of 1960' s that a systematic study of the properties of this kernel appears 6 ,21,22. e. 8) and, as a consequence: R(e' -+ 30 Part I - Chapter 1. e. 10) 3) Reciprocity; this is a subtler property that follows from the circumstance that the microscopic dynamics is time reversible and the wall is assumed to be in a local equilibrium state, not significantly disturbed by the impinging molecule.
18) is the difference between the impinging and emerging flow of the quantity, whose density is 'I/J, when the distribution is Mw4>; the denominator is the same thing when the restriction of f to e . n >0 is replaced by the wall Maxwellian, normalized in such a way as to give the same entering flow rate as f. In particular, if we let 'I/J = n, we obtain the accommodation coefficient for normal momentum, if we let 'I/J = t we obtain the accommodation coefficient for tangential momentum (in the direction of the unit vector t, tangent to the wall), if we let 'I/J = lel 2 , we obtain the accommodation coefficient for energy.