Green's function and pointwise estimate for a generalized Poisson‐Nernst‐Planck‐Navier‐Stokes model in dimension three

The Cauchy's problem of the generalized Poisson‐Nernst‐Planck‐Navier‐Stokes model in dimension three is considered. First, after dividing the physical domain into two parts: finite Mach number region and outside finite Mach number region, we give pointwise estimates of the Green's function by using long‐wave short‐wave decomposition and weighted energy estimate method in each region separately. Then from Duhamel's principle and some estimates on the nonlinear interactions between different wave patterns, we give the pointwise estimates of the solution for the nonlinear problem [Disp. Item 1.1. 1.1 ρt+ div (ρu)=0,(ρu)t+ div (ρu⊗u)+∇p=μΔu+μ′∇ div u−∇v−∇w+κΔϕ∇ϕ,vt+ div ...]–[Disp. Item 1.2. 1.2 (ρ,u,v,w)(x,t)|t=0=(ρ0,u0,v0,w0)(x),x∈R3, ...], which exhibit generalized Huygens' principle for mass density, macroscopic momentum, negative charge distribution and positive charge distribution. As a byproduct, we find that the decay rates inL p ( R 3 )( 2 < p ≤ ∞ ) for both negative charge distribution and positive charge distribution are faster than those for the mass density ρ and the macroscopic momentum. More importantly, we obtained both electrostatic potential and the difference between negative charge distribution and positive charge distribution have the exponential decay rate in L p ‐norm when p ≥ 1 .