### Abstract

This paper deals with the L_{p}-L_{q} decay estimate of the C^{0} analytic semigroup {T (t)}t≥0 associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier-Stokes equations in an exterior domain. We proved that ||δ^{j}T(t)f||_{L}_{p} ≤ C_{p,q}^{t} - j/2 - N/2(1/q - 1/p) ||f||_{L}_{q} (j = 0,1) provided that 1 < q ≤ p ≤ ∞ and q ≠ ∞. Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier-Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero.

Original language | English |
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Pages (from-to) | 33-72 |

Number of pages | 40 |

Journal | Asymptotic Analysis |

Volume | 107 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Keywords

- Exterior domains
- Free boundary problem
- L-L decay estimate
- Stokes equations
- Without surface tension

### ASJC Scopus subject areas

- Mathematics(all)