Global large solutions of the Cauchy problem for the NS-NPP equations with Fujita-Kato type initial data

Abstract

In this paper, we prove the global well-posedness of the Cauchy problem for the NS-NPP equations with some large Fujita-Kato type initial data. Specifically, we show that there exist two positive constants $c_{0}$ and $C_{0}$ such that if the initial data $(u_{0}, N_{0}, P_{0})$ satisfies the following condition: \begin{equation*} \left(\|u_0\|_{\dot{H}^{-1+\frac{d}{2}}}+\|N_{0}-P_{0}\|_{\dot{H}^{-2+\frac{d}{2}}}\right)\exp\left\{C_{0}\big(\|N_{0}+P_{0}\|_{\dot{H}^{-2+\frac{d}{2}}}^2+1\big)\right\} \leq c_{0}, \end{equation*} then the NS-NPP equations admits a unique global solution. This result implies global existence of solutions without any smallness conditions imposed on the sum of initial particle densities of negative and positive electric charge in the framework of Sobolev spaces.