Journal of Evolution Equations

, Volume 18, Issue 4, pp 1675–1696 | Cite as

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

  • Qiao Liu
  • Jihong ZhaoEmail author


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.


Navier–Stokes/Poisson–Nernst–Planck system Blowup Pressure Anisotropic Lebesgue spaces 

Mathematics Subject Classification

35B44 35K55 35Q35 76W05 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHunan Normal UniversityChangshaChina
  2. 2.School of Mathematics and Information ScienceBaoji University of Arts and ScienceBaojiChina
  3. 3.Institute of Applied Mathematics, College of ScienceNorthwest A&F UniversityYanglingChina

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