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


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Article Info

DATE

2018-12

AUTHORS

Qiao Liu, Jihong Zhao

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 ∫0TPLx1pLx2qLx3rβdt=+∞,where 2β+1p+1q+1r=2, 2≤p,q,r≤∞ and 1-(1p+1q+1r)≥0, and ∫0T∇PLx1pLx2qLx3rβdt=+∞,where 2β+1p+1q+1r=3, 1≤p,q,r≤∞ and 2-(1p+1q+1r)≥0, and ∫0T‖∂3P‖Lx3γLx1x2αβdt=+∞,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. More... »

PAGES

1-22

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http://dx.doi.org/10.1007/s00028-018-0456-0

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46 schema:description 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 ∫0TPLx1pLx2qLx3rβdt=+∞,where 2β+1p+1q+1r=2, 2≤p,q,r≤∞ and 1-(1p+1q+1r)≥0, and ∫0T∇PLx1pLx2qLx3rβdt=+∞,where 2β+1p+1q+1r=3, 1≤p,q,r≤∞ and 2-(1p+1q+1r)≥0, and ∫0T‖∂3P‖Lx3γLx1x2αβdt=+∞,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.
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