Global existence of large solutions for the three‐dimensional incompressible Navier–Stokes–Poisson–Nernst–Planck equations.

This work is concerned with the global existence of large solutions to the three‐dimensional dissipative fluid‐dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants c0,C0$$ {c}_0,{C}_0 $$ such that if the initial data satisfies u0hB˙p,1−1+3p+‖N0−P0‖B˙q,1−2+3qexpC0u03B˙p,1−1+3p2+‖N0+P0‖B˙r,1−2+3r+1expC0u03B˙p,1−1+3p+1≤c0,$$ {\displaystyle \begin{array}{ll}\left({\left\Vert {u}_0^h\right\Vert}_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}+{\left\Vert {N}_0-{P}_0\right\Vert}_{{\dot{B}}_{q,1}^{-2+\frac{3}{q}}}\right)& \exp \left\{{C}_0\left({\left\Vert {u}_0^3\right\Vert}_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}^2+\left({\left\Vert {N}_0+{P}_0\right\Vert}_{{\dot{B}}_{r,1}^{-2+\frac{3}{r}}}+1\right)\right.\right.\\ {}& \left.\left.\exp \left\{{C}_0{\left\Vert {u}_0^3\right\Vert}_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}\right\}+1\right)\right\}\le {c}_0,\end{array}} $$then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution. [ABSTRACT FROM AUTHOR]