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

Abstract

We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1 2 .Proposition 2.5. Let f be a suitable solution to the Landau equation (1) on [0, 1] satisfying (7) for some negligible set N ⊂ (0, 1], some q ∈ ( 6 5 , 2), and some C ′ E > 0.

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

“…Note that in order to give a rigorous proof of these facts, we apply the estimates obtained in this paper to solutions of an approximated problem and then pass to the limit. Finally, combining our result with the previous result in [14], we see that the set of singular times for weak solutions is included in a subset of the interval [T , T * ] whose Hausdorff dimension is at most 1/2. 1.4.4.…”

“…Proposition 1.3 describes a potential blowup phenomenon for solutions to the Landau equation with Coulomb potential. We recall that restrictions are given in [15] and [14] on the possible appearance of such a blowup. Our lower bound for the blowup rate is given in terms of relative entropy.…”

“…However, the lower bound is given in terms of relative entropy and thus probably cannot be used to estimate the size of singular times. We recall that the size of that set for the Landau equation with Coulomb potential is anyway estimated in [14].…”

“…• Partial regularity issue: Very recently Golse, Gualdani, Imbert and Vasseur [14] proved that the set of singular times for (suitable) weak solutions of the spatially homogeneous Landau equation with Coulomb potential has Hausdorff dimension at most 1/2 if the initial data possesses all polynomial moments. The key ingredient of the proof lies in the application of De Giorgi's method to a scaled suitable solution.…”

Well-posedness of the Lenard-Balescu equation with smooth interactions

“…We emphasize that the existence of global smooth solutions is an open question even in the Landau case, cf. [26,Chapter 5,§1.3(2)]; we refer to [11,8] for recent advances on the topic. For the Lenard-Balescu equation, getting beyond the above local-in-time result would even bring further difficulties due to the possible degeneracy of the dispersion function away from equilibrium.…”