Diffusive expansion for solutions of the Boltzmann equation in the whole space

This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space is an element of partial derivative(t) F-is an element of + v . del F-x(is an element of) = 1/is an element of Q (F-is an element of, F-is an element of), x is an element of R-N, v is an element of R-N, t > 0, (0.1) with prescribed initial data F-is an element of (t, x, v) vertical bar t=0 = F-0(is an element of) (x, v), x E R-N, v E R-N. (0.2) Our main purpose is to justify the global validity of the diffusive expansion F-is an element of (t, x, v) = mu + root mu{is an element of f(1) (t, x, v) + is an element of(2) f(2) (t, x, v) + ... + is an element of(n-1) f(n-1) (t, x, v) + is an element of(n) f(n)(is an element of) (t, x, v)} (0.3) for a solution F-is an element of (t, x, v) of the rescaled Boltzmann equation (0.1) in the whole space R-N for all t >= 0 with initial data F-0(is an element of)(x, v) satisfying the initial expansion F-0(is an element of)(x, v) = mu + root mu {is an element of f(1) (0, x, v) + is an element of(2) f(2)(0, x, v) + ... + is an element of(n-1) f(n-1) (0, x, v) + is an element of(n) f(n)(is an element of) (0, x, V)}, (X, V) is an element of R-2N. (0.4) Here mu(v) = (2 pi)-N/2 exp(-[v](2)/2) is a normalized global Maxwellian. Under the assumption that the fluid components of the coefficients f(m)(0, x, v) (1 <= m <= n) of the initial expansion F-0(is an element of) (x, v) have divergence-free velocity fields u(m)(0)(x) as well as temperature fields theta(0)(m)(x), if we assume further that the velocity-temperature fields [u(1)(0)(x), theta(0)(1)(x)] of f(1)(0, x, v) have small amplitude in H-s(R-N)(s >= 2(N+ n + 2)), we can determine these coefficients f(m)(t, x, v) (1 <= m <= n) in the diffusive expansion (0.3) uniquely by an iteration method and energy method. The hydrodynamic component of these coefficients fm(t, x, v) (1 <= m <= n) satisfies the incompressible condition, the Boussinesq relations and/or the Navier-Stokes-Fourier system respectively, while the microscopic component of these coefficients is determined by a recursive formula. Compared with the corresponding problem inside a periodic box studied in Y. Guo (2006) 1181, the main difficulty here is due to the fact that Poincare's inequality is not valid in the whole space RN and this difficulty is overcome by using the L-p-L-q-estimate on the Riesz potential. Moreover, by exploiting the energy method, we can also deduce certain the space-time energy estimates on these coefficients f(m)(t, x, v) (1 <= m <= n). Once the coefficients f(m)(t, x, v) (1 <= m <= n) in the diffusive expansion (0.3) are uniquely determined and some delicate estimates have been obtained, the uniform estimates with respect to is an element of on the remainders f(n)(is an element of) (t, x, v) are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion (0.3) in the large provided that N > 2n + 2 (0.5) Notice that for m >= 2, u(m)(t, x) is no longer a divergence-free vector and it is worth to pointing out that, for m >= 3, it was in deducing certain estimates on p(m)(t, x) by the L-P-L-q-estimate on the Riesz potential that we need to require that N > 2n + 2. (C) 2010 Elsevier Inc. All rights reserved.