### Abstract

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the L_{p} in time and the L_{q} in space framework with 2 <p <1 and N <q <∞ under the assumption that the initial domain is a uniform W_{q} ^{2-1/q} domain in ℝ^{N}(N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal L_{p}-L_{q} regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal L_{p}-L_{q} regularity theorem.

Original language | English |
---|---|

Pages (from-to) | 3741-3774 |

Number of pages | 34 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 36 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2016 Jul 1 |

### Fingerprint

### Keywords

- Compressible viscous fluid
- Free boundary problem
- Local well-posedness theorem
- Maximal L-L regularity
- R-bounded solution operator
- Two phase problem
- Uniform W domain

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

*Discrete and Continuous Dynamical Systems*,

*36*(7), 3741-3774. https://doi.org/10.3934/dcds.2016.36.3741