THE GHOST EFFECT IN THE CONTINUUM LIMIT FOR A VAPOR–GAS MIXTURE AROUND CONDENSED PHASES: ASYMPTOTIC ANALYSIS OF THE BOLTZMANN EQUATION

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• From Eqs. 3.18 and 3.21, it is seen that Eq. 3.32 is equivalent to a relation among the boundary values of u iH1 ni , , and (∂X H0 A /∂xi )ni. Therefore, if we regard Eqs. 3.15, 3.16, 3.17 with Eqs. 3.18, 3.19, 3.20, 3.21, 3.22 as the equations for , and [see the last part of the paragraph containing Eq. 3.22], then the boundary conditions 3.25a, 3.25b, 3.32, and 3.33 are interpreted as follows: Eq. 3.25a, 3.25b specifies the boundary values of and ; Eq. 3.32 gives a relation among the boundary values of u iH1 ni , , and ; and Eq. 3.33 gives a relation among the boundary values of u iH1 ti , , and
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• In fact, if the assumptions 3.11 and 3.23 are not compatible with the physical problem under consideration, the analysis cannot be performed consistently. The compatibility does not exclude the possibility of a solution with a flow of O(1), but the presence of such a solution is unlikely in the situation without external forces
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• As discussed in Ref. [1], the factor I has a different nature from others. The boundary data and of the system in the continuum limit are well defined; therefore, if we assume that Ui = 0, all the effects caused by the factors II–VII are determined unambiguously. In contrast, the boundary data Ui , on which the factor I is based, cannot be known from the system in the continuum limit. In other words, the effect caused by the factor I cannot be determined only by the information from the system that is in the continuum limit. This fact shows that fatal ambiguity is also contained in the classical fluid dynamics
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