### Abstract

In this paper, we prove a local in time unique existence theorem for the freeboundary problem of a compressible barotropic viscous fluid flow without surface tensionin the L_{p} in time and L_{q} in space framework with 2 < p < ∞ and N < q < ∞ under the assumption that the initial domain is a uniform W^{2-1/q}_{q} one in ℝ^{N} (N ≥ 2).After transforming a unknown time dependent domain to the initial domain by theLagrangian transformation, we solve problem by the Banach contraction mappingprinciple based on the maximal L_{p}-L_{q} regularity of the generalized Stokes operatorfor the compressible viscous fluid flowwith free boundary condition. The key issue forthe linear theorem is the existence of R-bounded solution operator in a sector, whichcombined with Weis's operator valued Fourier multiplier theorem implies the generationof analytic semigroup and the maximal L_{p}-L_{q} regularity theorem. The nonlinearproblem we studied here was already investigated by several authors (Denisova andSolonnikov, St. Petersburg Math J 14:1-22, 2003; J Math Sci 115:2753-2765, 2003; Secchi, Commun PDE 1:185-204, 1990; Math Method Appl Sci 13:391-404, 1990;Secchi and Valli, J Reine Angew Math 341:1-31, 1983; Solonnikov and Tani, Constantincarathéodory: an international tribute, vols 1, 2, pp 1270-1303,World ScientificPublishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin,1992; Tani, J Math Kyoto Univ 21:839-859, 1981; Zajaczkowski, SIAM JMath Anal25:1-84, 1994) in the L2 framework and Hölder spaces, but our approach is differentfrom them.

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

Pages (from-to) | 55-89 |

Number of pages | 35 |

Journal | Annali dell'Universita di Ferrara |

Volume | 60 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

### Keywords

- Compressible viscous fluid
- Free boundary problem
- Local in time existence theorem
- R-bounded solution operator

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Annali dell'Universita di Ferrara*,

*60*(1), 55-89. https://doi.org/10.1007/s11565-013-0194-8

**On some free boundary problem for a compressible barotropic viscous fluid flow.** / Enomoto, Yuko; von Below, Lorenz; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

*Annali dell'Universita di Ferrara*, vol. 60, no. 1, pp. 55-89. https://doi.org/10.1007/s11565-013-0194-8

}

TY - JOUR

T1 - On some free boundary problem for a compressible barotropic viscous fluid flow

AU - Enomoto, Yuko

AU - von Below, Lorenz

AU - Shibata, Yoshihiro

PY - 2014

Y1 - 2014

N2 - In this paper, we prove a local in time unique existence theorem for the freeboundary problem of a compressible barotropic viscous fluid flow without surface tensionin the Lp in time and Lq in space framework with 2 < p < ∞ and N < q < ∞ under the assumption that the initial domain is a uniform W2-1/qq one in ℝN (N ≥ 2).After transforming a unknown time dependent domain to the initial domain by theLagrangian transformation, we solve problem by the Banach contraction mappingprinciple based on the maximal Lp-Lq regularity of the generalized Stokes operatorfor the compressible viscous fluid flowwith free boundary condition. The key issue forthe linear theorem is the existence of R-bounded solution operator in a sector, whichcombined with Weis's operator valued Fourier multiplier theorem implies the generationof analytic semigroup and the maximal Lp-Lq regularity theorem. The nonlinearproblem we studied here was already investigated by several authors (Denisova andSolonnikov, St. Petersburg Math J 14:1-22, 2003; J Math Sci 115:2753-2765, 2003; Secchi, Commun PDE 1:185-204, 1990; Math Method Appl Sci 13:391-404, 1990;Secchi and Valli, J Reine Angew Math 341:1-31, 1983; Solonnikov and Tani, Constantincarathéodory: an international tribute, vols 1, 2, pp 1270-1303,World ScientificPublishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin,1992; Tani, J Math Kyoto Univ 21:839-859, 1981; Zajaczkowski, SIAM JMath Anal25:1-84, 1994) in the L2 framework and Hölder spaces, but our approach is differentfrom them.

AB - In this paper, we prove a local in time unique existence theorem for the freeboundary problem of a compressible barotropic viscous fluid flow without surface tensionin the Lp in time and Lq in space framework with 2 < p < ∞ and N < q < ∞ under the assumption that the initial domain is a uniform W2-1/qq one in ℝN (N ≥ 2).After transforming a unknown time dependent domain to the initial domain by theLagrangian transformation, we solve problem by the Banach contraction mappingprinciple based on the maximal Lp-Lq regularity of the generalized Stokes operatorfor the compressible viscous fluid flowwith free boundary condition. The key issue forthe linear theorem is the existence of R-bounded solution operator in a sector, whichcombined with Weis's operator valued Fourier multiplier theorem implies the generationof analytic semigroup and the maximal Lp-Lq regularity theorem. The nonlinearproblem we studied here was already investigated by several authors (Denisova andSolonnikov, St. Petersburg Math J 14:1-22, 2003; J Math Sci 115:2753-2765, 2003; Secchi, Commun PDE 1:185-204, 1990; Math Method Appl Sci 13:391-404, 1990;Secchi and Valli, J Reine Angew Math 341:1-31, 1983; Solonnikov and Tani, Constantincarathéodory: an international tribute, vols 1, 2, pp 1270-1303,World ScientificPublishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin,1992; Tani, J Math Kyoto Univ 21:839-859, 1981; Zajaczkowski, SIAM JMath Anal25:1-84, 1994) in the L2 framework and Hölder spaces, but our approach is differentfrom them.

KW - Compressible viscous fluid

KW - Free boundary problem

KW - Local in time existence theorem

KW - R-bounded solution operator

UR - http://www.scopus.com/inward/record.url?scp=84899840501&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899840501&partnerID=8YFLogxK

U2 - 10.1007/s11565-013-0194-8

DO - 10.1007/s11565-013-0194-8

M3 - Article

AN - SCOPUS:84899840501

VL - 60

SP - 55

EP - 89

JO - Annali dell'Universita di Ferrara

JF - Annali dell'Universita di Ferrara

SN - 0430-3202

IS - 1

ER -