Blowup criteria in terms of pressure for the 3D nonlinear dissipative system modeling electro-diffusion

Abstract

In this paper, we consider some sufficient conditions for the breakdown of local smooth solutions to the Cauchy problem of the 3D Navier–Stokes/Poisson–Nernst–Planck system modeling electro-diffusion in terms of pressure (or gradient of pressure or one directional derivative of pressure) in the framework of the anisotropic Lebesgue spaces. Precisely, let T be the maximum existence time of local smooth solution. Then if T< + ∞, we have ∫0T∥∥∥P∥Lx1p∥Lx2q∥Lx3rβdt=+∞,where 2β+1p+1q+1r=2, 2 ≤ p, q, r≤ ∞ and 1-(1p+1q+1r)≥0, and ∫0T∥∥∥∇P∥Lx1p∥Lx2q∥Lx3rβdt=+∞,where 2β+1p+1q+1r=3, 1 ≤ p, q, r≤ ∞ and 2-(1p+1q+1r)≥0, and ∫0T∥‖∂3P‖Lx3γ∥Lx1x2αβdt=+∞,where 2β+1γ+2α=k∈[2,3) and 3k≤γ≤α<1k-2. These results are even new for the 3D incompressible Navier–Stokes equations.