C^\infty smoothing for weak solutions of the inhomogeneous Landau equation

Partial Regularity in Time for the Space Homogeneous Landau Equation with Coulomb Potential

“…Of course, the existence and uniqueness theory for the space inhomogeneous Landau equation is even harder, and most of the existing results on that equation bear on near-Maxwellian equilibrium global solutions [18,6,8], apart from the very general weak stability result in [28]. There are also various local existence and uniqueness results, as well as smoothing estimates for solutions of the space inhomogeneous Landau equation under the assumption of locally (in time and space) bounded moments in v: see [19,21,20] (notice that [21,20] require only that the distribution function has bounded moments in v of order larger than 9 2). Since the present paper is focused on the Coulomb case, which is the most interesting on physical grounds, we have omitted the rather large literature on the generalizations of the Landau equation where the collision kernel a is replaced with z γ+2 Π(z) with γ > −3.…”

Gaussian lower bounds for the Boltzmann equation without cut-off

“…Previous results of conditional regularity include, for the Boltzmann equation and under the conditions above, the proof of L ∞ bounds in [26], the proof of a weak Harnack inequality and Hölder continuity in [19], the proof of polynomially decaying upper bounds in [18], and the proof of Schauder estimates to bootstrap higher regularity estimates in [20]. In the case of the closely related Landau equation, which is a nonlinear diffusive approximation of the Boltzmann equation, the L ∞ bound was proved in [27,13], the Harnack inequality and Hölder continuity were obtained in [28,13], decay estimates were obtained in [10], and Schauder estimates were established in [16] (see also [17]). The interested reader is refereed to the short review [23] of the conditional regularity program.…”

A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications

“…In the spatially inhomogeneous context, we quote [38] for the existence of renormalized solutions and [17] and [23] for the global well-posedness near Maxwellian and the local well-posedness in weighted Sobolev spaces. We finally refer to [3] for a general perturbation result, and to [24], [25] for conditional regularity results.…”