### Abstract

In a bounded smooth domain Ω ⸦ ℝ^{3} and a time interval [0, T), 0 < T ≤ ∞, consider the instationary Navier-Stokes equations with initial value U_{0} ∈ L^{2} _{σ}(Ω) and external force f = divF, F ∈ L^{2}(0, T;L^{2}(Ω)). As is well known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when U|_{∂Ω}= g with non-zero time-dependent boundary values g. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.

Original language | English |
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Title of host publication | Progress in Nonlinear Differential Equations and Their Application |

Publisher | Springer US |

Pages | 211-232 |

Number of pages | 22 |

DOIs | |

Publication status | Published - 2011 Jan 1 |

### Publication series

Name | Progress in Nonlinear Differential Equations and Their Application |
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Volume | 80 |

ISSN (Print) | 1421-1750 |

ISSN (Electronic) | 2374-0280 |

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### Keywords

- Energy inequality
- Instationary Navier-Stokes equations
- Long-time behavior
- Non-zero boundary values
- Time-dependent data
- Weak solutions

### ASJC Scopus subject areas

- Analysis
- Computational Mechanics
- Mathematical Physics
- Control and Optimization
- Applied Mathematics

### Cite this

*Progress in Nonlinear Differential Equations and Their Application*(pp. 211-232). (Progress in Nonlinear Differential Equations and Their Application; Vol. 80). Springer US. https://doi.org/10.1007/978-3-0348-0075-4_11