TY  - JOUR
AU  - Kim, Jinoh
AU  - Guo, Yan
AU  - Hwang, Hyung Ju
PY  - 2020
TI  - An  L2 to  L∞ Framework for the Landau Equation
JO  - Peking Mathematical Journal
SN  - 2096-6075
SP  - 131
EP  - 202
VL  - 3
IS  - 2
AB  - Consider the Landau equation with Coulomb potential in a periodic box. We develop a new  L2 to L∞ framework to construct global unique solutions near Maxwellian with small  L∞ norm. The first step is to establish global  L2 estimates with strong velocity weight and time decay, under the assumption of  L∞ bound, which is further controlled by such  L2 estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv:1505.04608 (2015)). The second step is to employ estimates in  Sp spaces to control velocity derivatives to ensure uniqueness, which is based on Hölder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv:1506.01908 (2015), Imbert and Mouhot in arXiv:1505.04608 (2015)).
UR  - https://doi.org/10.1007/s42543-019-00018-x
DO  - 10.1007/s42543-019-00018-x