TY - JOUR

T1 - Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

AU - Farwig, R.

AU - Kozono, H.

AU - Sohr, H.

PY - 2011

Y1 - 2011

N2 - Consider a smooth bounded domain Ω ⊆ ℝ3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u0 ∈ L2σ(Ω) and external force f = div F, F ∈ L2(0, T;L2(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 ⊆ ℝ3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u0 ∈ L2σ(Ω) and external force f = div F, F ∈ L2(0, T;L2(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.

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U2 - 10.4171/RSMUP/125-4

DO - 10.4171/RSMUP/125-4

M3 - Article

AN - SCOPUS:84856151283

SN - 0041-8994

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

ER -