A coupling approach for the convergence to equilibrium for a collisionless gas

Convergence toward the steady state of a collisionless gas with Cercignani–Lampis boundary condition

“…The conclusion from Corollary 23 is that the rate of convergence towards the steady state of the free-transport equation with Cercignani-Lampis boundary condition is better than 1 t d (up to a log factor) when starting from an initial datum with enough regularity. As this is also the rate obtained for the pure diffuse boundary condition (see for instance [31] for the spherically symmetric case, and [4], [7] for the general case), which corresponds to the particular case r ⊥ = r = 1, and since it is known that this rate is optimal in this context, we can conclude to the optimality for the general Cercignani-Lampis boundary condition.…”

“…Ultimately their results allow one to handle the Maxwell boundary condition with various temperatures at the boundary. Another probabilistic approach was taken by Bernou and Fournier [7] through the use of a probabilistic coupling, based on a description of the problem with a stochastic process. This allowed the authors to conclude to the optimal rate 1 t d in the general case of a C 2 regular domain, with constant temperature.…”

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

“…For the diffuse reflection (20), they identify the lack of uniform convergence for L 1 initial data, and, under some additional regularity, proved a convergence rate of (1 + t) −1 , in the L 1 distance. This result was improved up to the optimal rate 1 (1+t) d− in several subsequent articles by Kuo, Liu and Tsai [48,49] and Kuo [47] in a radial domain, and by Bernou-Fournier [11] and Bernou [8] in a C 2 bounded one, and ultimately culminated in a treatment of the more general Cercignani-Lampis boundary condition [9], for which the same rate of convergence was obtained. The key outcome of those research is that stochastic boundary conditions ((MBC) and (CLBC)) provide only a polynomial rate of convergence in the L 1 distance: there is no spectral gap for those dynamics.…”

“…In the first setting, only Hypotheses 1 and 2 are assumed. We prove that the rate of convergence is then bounded from above by the (optimal) polynomial rate of (1 + t) −d derived for the free-transport equation in [8,9,11]. In the second framework, σ is almost everywhere bounded from below by a positive constant, and exponential convergence towards the steady state is derived.…”

“…The speed of convergence can thus be read at this level. This strategy is in some sense analogous to a probabilistic coupling -one such for the free-transport dynamics is performed in [11] -but relying on the framework introduced by Hairer-Mattingly [39,40] and refined by Cañizo-Mischler [17] allows to escape the corresponding cumbersome construction -especially in models like the ones investigated here, whose probabilistic writing would involve several sources of randomness -by playing with weighted norms instead. We obtain the existence and uniqueness of the steady state, and some rate of convergence towards it.…”