## Abstract

We derive a hydrodynamic model of the compressible conductive fluid by using an energetic variational approach, which could be called a generalized Poisson-Nernst-Planck-Navier-Stokes system. This system characterizes the micro-macro interactions of the charged fluid and the mutual friction between the crowded charged particles. In particular, it reveals the cross-diffusion phenomenon which does not happen in the fluid with the dilute charged particles. The cross-diffusion is tricky; however, we develop a general method to show that the system is globally asymptotically stable under small perturbations around a constant equilibrium state. Under some conditions, we also obtain the optimal decay rates of the solution and its derivatives of any order. Our method will apply equally well to a class of cross-diffusion systems if their linearized diffusion matrices are diagonally dominant.

Original language | English (US) |
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Pages (from-to) | 3191-3235 |

Number of pages | 45 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 48 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2016 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics