TY - JOUR

T1 - On the stationary Navier-Stokes equations in exterior domains

AU - Kim, Hyunseok

AU - Kozono, Hideo

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

PY - 2012/11/15

Y1 - 2012/11/15

N2 - This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L 3/2,∞(Ω)∩L p,q(Ω) with 3/2p,q(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L 3/2,∞(Ω)∩L 2(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L 3/2(Ω)+L 2(Ω), which generalizes Leray's existence result.

AB - This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L 3/2,∞(Ω)∩L p,q(Ω) with 3/2p,q(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L 3/2,∞(Ω)∩L 2(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L 3/2(Ω)+L 2(Ω), which generalizes Leray's existence result.

KW - Energy inequality

KW - Exterior problem

KW - Lorentz space

KW - Navier-Stokes equations

KW - Regularity

KW - Uniqueness

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U2 - 10.1016/j.jmaa.2012.05.039

DO - 10.1016/j.jmaa.2012.05.039

M3 - Article

AN - SCOPUS:84864016579

VL - 395

SP - 486

EP - 495

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -