Consider the Landau equation with Coulomb potential in a periodic box.We develop a new L^(2)to L^(∞)framework to construct global unique solutions near Maxwellian with small L^(∞)norm.The first step is to establish ...Consider the Landau equation with Coulomb potential in a periodic box.We develop a new L^(2)to L^(∞)framework to construct global unique solutions near Maxwellian with small L^(∞)norm.The first step is to establish global L^(2)estimates with strong velocity weight and time decay,under the assumption of L^(∞)bound,which is further controlled by such L^(2)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 S_(p)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 Vas-seur in arXiv:1506.01908(2015),Imbert and Mouhot in arXiv:1505.04608(2015)).展开更多
基金Yan Guo is supported in part by NSF grant#DMS-1810868Chinese NSF Grant#10828103as well as a Simon Fellowship.Hyung Ju Hwang was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF-2017R1E1A1A03070105,NRF-2019R1A5A1028324).
文摘Consider the Landau equation with Coulomb potential in a periodic box.We develop a new L^(2)to L^(∞)framework to construct global unique solutions near Maxwellian with small L^(∞)norm.The first step is to establish global L^(2)estimates with strong velocity weight and time decay,under the assumption of L^(∞)bound,which is further controlled by such L^(2)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 S_(p)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 Vas-seur in arXiv:1506.01908(2015),Imbert and Mouhot in arXiv:1505.04608(2015)).
