THE 3D NONLINEAR DISSIPATIVE SYSTEM MODELING ELECTRO-DIFFUSION WITH BLOW-UP IN ONE DIRECTION

This paper establishes a sufficient condition for the breakdown of local smooth solutions, to the Cauchy problem of the 3D Navier-Stokes/Poisson-Nernst-Planck system modeling electro-diffusion, via one directional derivative of the horizontal component of the velocity field (i.e., (partial derivative(i)u(1),partial derivative(j)u(2),0) where i, j is an element of {1,2,3}) in the framework of the anisotropic Lebesgue spaces. More precisely, let T-* > 0 be the finite and maximum existence time of local smooth solution. Then integral(T*)(0)(parallel to parallel to partial derivative(i)u(1)(t)parallel to L-xi(alpha) parallel to(q)(Lx (i) over capx (i) over tilde beta) + parallel to parallel to partial derivative(j)u(2)(t)parallel to L-xj(alpha) parallel to(q)(Lx (i) over capx (i) over tilde beta)) dt = +infinity, with 2/q + 1/alpha + 2/beta = m is an element of [1, 3/2) and 3/m < alpha <= beta <= 1/m-1, where (i,<(i)over cap>,(i) over tilde) and (j,(j) over cap,(j) over tilde) belong to the permutation group on the set S-3 := {1,2,3}. This reveals that the horizontal component of the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system.