Abstract
Consider a smooth bounded domain Ω ⊆ ℝ 3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u 0 ∈ L 2 σ(Ω) and external force f = div F, F ∈ L 2(0, T;L 2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} ∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe} ∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.
| Original language | English |
|---|---|
| Pages (from-to) | 51-70 |
| Number of pages | 20 |
| Journal | Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova |
| Volume | 125 |
| Publication status | Published - 2011 |
| Externally published | Yes |
Fingerprint
ASJC Scopus subject areas
- Algebra and Number Theory
- Analysis
- Geometry and Topology
- Mathematical Physics
Cite this
Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence. / Farwig, R.; Kozono, Hideo; Sohr, H.
In: Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova, Vol. 125, 2011, p. 51-70.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence
AU - Farwig, R.
AU - Kozono, Hideo
AU - Sohr, H.
PY - 2011
Y1 - 2011
N2 - Consider a smooth bounded domain Ω ⊆ ℝ 3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u 0 ∈ L 2 σ(Ω) and external force f = div F, F ∈ L 2(0, T;L 2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} ∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe} ∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.
AB - Consider a smooth bounded domain Ω ⊆ ℝ 3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u 0 ∈ L 2 σ(Ω) and external force f = div F, F ∈ L 2(0, T;L 2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} ∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe} ∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.
UR - http://www.scopus.com/inward/record.url?scp=84856151283&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84856151283&partnerID=8YFLogxK
M3 - Article
VL - 125
SP - 51
EP - 70
JO - Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova
JF - Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova
SN - 0041-8994
ER -