TY - JOUR

T1 - Weak solutions of the stationary Navier-Stokes equations for a viscous incompressible fluid past an obstacle

AU - Heck, Horst

AU - Kim, Hyunseok

AU - Kozono, Hideo

N1 - Funding Information:
The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MEST) (No. 2010-0002536).

PY - 2013/6

Y1 - 2013/6

N2 - Consider the stationary Navier-Stokes equations in an exterior domain Ω ⊂ ℝ3 with smooth boundary. For every prescribed constant vector u∞ ≠ 0 and every external force f ∈ Ḣ2-1(Ω), Leray (J. Math. Pures. Appl., 9:1-82, 1933) constructed a weak solution u with ∇u ∈ L2(Ω) and u-u∞ ∈ L6(Ω). Here Ḣ2-1(Ω) denotes the dual space of the homogeneous Sobolev space Ḣ12(Ω). We prove that the weak solution u fulfills the additional regularity property u-u∞ ∈ L4(Ω) and u∞ · ∇u Ḣ2-(Ω) without any restriction on f except for f ∈ Ḣ2-1(Ω). As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that {double pipe}f{double pipe}Ḣ-12(Ω) and {pipe} u∞{pipe} are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1-82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as u∞→0 in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case u∞≠0.

AB - Consider the stationary Navier-Stokes equations in an exterior domain Ω ⊂ ℝ3 with smooth boundary. For every prescribed constant vector u∞ ≠ 0 and every external force f ∈ Ḣ2-1(Ω), Leray (J. Math. Pures. Appl., 9:1-82, 1933) constructed a weak solution u with ∇u ∈ L2(Ω) and u-u∞ ∈ L6(Ω). Here Ḣ2-1(Ω) denotes the dual space of the homogeneous Sobolev space Ḣ12(Ω). We prove that the weak solution u fulfills the additional regularity property u-u∞ ∈ L4(Ω) and u∞ · ∇u Ḣ2-(Ω) without any restriction on f except for f ∈ Ḣ2-1(Ω). As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that {double pipe}f{double pipe}Ḣ-12(Ω) and {pipe} u∞{pipe} are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1-82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as u∞→0 in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case u∞≠0.

UR - http://www.scopus.com/inward/record.url?scp=84877109339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877109339&partnerID=8YFLogxK

U2 - 10.1007/s00208-012-0861-6

DO - 10.1007/s00208-012-0861-6

M3 - Article

AN - SCOPUS:84877109339

SN - 0025-5831

VL - 356

SP - 653

EP - 681

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 2

ER -