Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

作者：Esposito R^*; Guo Y; Kim C; Marra R

来源：Communications in Mathematical Physics, 2013, 323(1): 177-239.

DOI：10.1007/s00220-013-1766-2

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem: nu center dot del F-x = 1/K(n)Q(F, F), (x, nu) is an element of Omega x R-3 F( x, v)vertical bar(n)(x) center dot v %26lt; 0 = mu theta integral(n( x) center dot v%26apos; %26gt; 0) F( x, v%26apos;)(n( x) center dot v%26apos;) dv%26apos;, x is an element of partial derivative Omega, (0.2) where Omega is a bounded domain in R-d, 1 %26lt;= d %26lt;= 3, Kn is the Knudsen number and mu theta = 1/2 pi theta(2)(x) exp[-vertical bar nu vertical bar(2)/2 theta(x)] is a Maxwellian with non-constant(non-isothermal) wall temperature.( x). Based on new constructive coercivity estimates for both steady and dynamic cases, for vertical bar theta - theta(0)vertical bar %26lt;= delta %26lt;%26lt; 1 = and any fixed value of K-n, we construct a unique non- negative solution F-s to (0.1) and (0.2), continuous away from the grazing set and exponentially asymptotically stable. This solution is a genuine non- equilibrium stationary solution differing from a local equilibrium Maxwellian. As an application of our results we establish the expansion Fs = mu(theta 0) + delta F-1 + O(delta(2)) and we prove that, if the Fourier law holds, the temperature contribution associated to F-1 must be linear, in the slab geometry.

• 出版日期2013-10