## Sharp regularity estimates for the spatially homogeneous Landau equation with Coulomb potential

Time: 2024-03-05
Published By: Fei Tao

**Speaker(s): ** Jie Ji(BICMR)

**Time: ** 15:00-16:00 March 11, 2024

**Venue: ** Room 29, Quan Zhai, BICMR

Abstract: In our current study, we aim to examine the precise regularity estimates for the solution $f = f(t, v)$ of the spatially homogeneous Landau-Coulomb equation. Specifically, we focus on the scenario where the initial data $f_0$ belongs to a weighted $L^{p}$ space with $p \geq 3/2$. Our results are summarized as follows:

(i). If the initial data $f_0$ possesses only polynomial moments, we establish the existence of a global or local solution. Moreover, within the relevant time span, the solution $f(t)$ exhibits the sharp smoothing estimates in {\it any Sobolev spaces} but with appropriate {\it negative weights}, which are similar to those for the linear heat equation:

$\|e^{-t\Delta}f_0\|_{H^n} \leq t^{-\frac{n}{2}-\frac{3}{2p}+\frac{3}{4}}\|f_0\|_{L^p}$.

Furthermore, we substantiate the necessity of these {\it negative weights}(in smoothing estimates) by constructing typical initial data.

(ii). We establish the propagation of exponential moments in $L^2$ space without any loss. Consequently, the solution belongs to the Gevrey class, exhibiting an optimal index that depends on the exponential moment for any positive time.

(i). If the initial data $f_0$ possesses only polynomial moments, we establish the existence of a global or local solution. Moreover, within the relevant time span, the solution $f(t)$ exhibits the sharp smoothing estimates in {\it any Sobolev spaces} but with appropriate {\it negative weights}, which are similar to those for the linear heat equation:

$\|e^{-t\Delta}f_0\|_{H^n} \leq t^{-\frac{n}{2}-\frac{3}{2p}+\frac{3}{4}}\|f_0\|_{L^p}$.

Furthermore, we substantiate the necessity of these {\it negative weights}(in smoothing estimates) by constructing typical initial data.

(ii). We establish the propagation of exponential moments in $L^2$ space without any loss. Consequently, the solution belongs to the Gevrey class, exhibiting an optimal index that depends on the exponential moment for any positive time.