Stability of the Maxwell–Stefan System in the Diffusion Asymptotics of the Boltzmann Multi-species Equation

From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations

“…The article by Bondesan and Briant [8] studies the Maxwell-Stefan asymptotics, in the 3-dimensional torus T 3 , of the system of coupled Boltzmann equations for mixtures (5)-( 6), with collisional integrals of the form (8)-( 9) and suitable cross sections (of cutoff Maxwellian type, or hard potentials, or hard spheres). The starting point is the Cauchy theory, proved by the same authors in [7], whose precise result is the following.…”

“…Briant and Grec have studied in [15] the asymptotics of the Boltzmann system (5)-( 6) in a framework different from the one studied in [8]. They suppose that, at the leading order, the species velocities are identical and obtain a cross-diffusion system of Fick type, not equivalent to the Maxwell-Stefan systems because the two matrices linking fluxes and concentrations are not invertible.…”

“…The scaling has been introduced in [13], and subsequently developed in [11] and [23]. After describing these formal results, we summarize the result by Bondesan and Briant [8], which provides a rigorous description of the limiting procedure. The next chapter is devoted to the reactive case, and is essentially based on the recent articles [2], [3] and [4].…”

Energy method for the Boltzmann equation of monatomic gaseous mixtures

“…Many works focused on the analysis of the monatomic case, like [6,12,9,3], which were dedicated to compactness, hypocoercivity-related and stability results. Well-posedness and regularity were investigated in [9,8,18,13], and asymptotics questions were tackled in [7,22,5,4]. Some of the previous papers, for instance [9], rely on the so-called micro-macro decomposition.…”

Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

“…Besides, the Maxwell-Stefan equations also raise a great theoretical interest, as their mathematical analysis appears to be very challenging. The difficulties come from the fact that summing over i the relations (2), we obtain a linear dependence on the mole fractions' gradients which imposes to introduce a further condition in order to close the system and provide a satisfactory Cauchy theory to (1)- (2). To our knowledge, the existing mathematical results that deal with the problem of existence and uniqueness of solutions to the sole system (1)-( 2) are all tied up to the assumption that the mixture is subject to a transient equimolar diffusion [24], namely the total diffusive flux satisfies N i=1 F i (t, x) = 0, t ≥ 0, x ∈ T 3 .…”