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 $T<+\infty$, we have

$$\begin{aligned} \int _{0}^{T}\left\| \left\| \left\| P\right\| _{L^{p}_{x_{1}}} \right\| _{L^{q}_{x_{2}}} \right\| _{L^{r}_{x_{3}}}^{\beta }\text {d}t=+\infty , \end{aligned}$$

where $\frac{2}{\beta }+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$, $2\le p,q,r\le \infty$ and $1-(\frac{1}{p}+\frac{1}{q}+\frac{1}{r})\ge 0$, and

$$\begin{aligned} \int _{0}^{T}\left\| \left\| \left\| \nabla P\right\| _{L^{p}_{x_{1}}}\right\| _{L^{q}_{x_{2}}} \right\| _{L^{r}_{x_{3}}}^{\beta }\text {d}t=+\infty , \end{aligned}$$

where $\frac{2}{\beta }+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=3$, $1\le p,q,r\le \infty$ and $2-(\frac{1}{p}+\frac{1}{q}+\frac{1}{r})\ge 0$, and

$$\begin{aligned} \int _{0}^{T}\left\| \Vert \partial _{3}P\Vert _{L^{\gamma }_{x_{3}}} \right\| _{L^{\alpha }_{x_{1}x_{2}}}^{\beta }\text {d}t=+\infty , \end{aligned}$$

where $\frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3)$ and $\frac{3}{k}\le \gamma \le \alpha < \frac{1}{k-2}$. These results are even new for the 3D incompressible Navier–Stokes equations.