2012/12/16 by Zhigang Wu, Weike Wang, Wu, Zhigang +1
Mathematics · #35B40 #35Q35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #math.AP #msc:35B40 #msc:35Q35
paper · pdf · doi:10.48550/arxiv.1212.3754
20 pages
openalex publication_date 2012/12/16 · arxiv created 2012/12/18 · arxiv updated 2012/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in ℝ3 when H3 norm of the initial data is small. If further, the H-s norm (0≤ s<3/2) or B2,∞-s norm (0<s≤3/2) of the initial data is bounded, we give the optimal decay rates of the solution. As a byproduct, the decay results of the Lp-L2 (1≤ p≤2) type hold without the smallness of the Lp norm of the initial data. In particular, we deduce that ‖∇k(ρ1-ρ2)‖L2 ∼(1+t)-5/4-(k)/(2) and ‖∇k(ρi-ρ,ui,∇ϕ)‖L2 ∼(1+t)-3/4-(k)/(2). We improve the decay results in Li and Yang \citeLi3(J.Differential Equations 252(2012), 768-791), where they showed the decay rates as ‖∇k(ρi-ρ)‖L2 ∼(1+t)-3/4-(k)/(2) and ‖∇k(ui,∇ϕ)‖L2 ∼(1+t)-1/4-(k)/(2), when the H3∩ L1 norm of the initial data is small. Our analysis is motivated by the technique developed recently in Guo and Wang \citeGuo(Comm. Partial Differential Equations 37(2012), 2165-2208) with some modifications.