On the two-dimensional steady-state problem of a viscous gas in an exterior domain

Abstract

We investigate plane steady flows of a viscous, isothermal, compressible fluid past an obstacle, with nonzero velocity at infinity. Using the decomposition of the velocity field onto its compressible and incompressible parts, we prove an existence and uniqueness theorem, under the assumption that the external data are "sufficiently small".

Liouville-type theorems for the stationary MHD equations in 2D

“…The above results also can be generalized to the shear thickening flows, for example see [3,4,5,18,11,19]. The existence and asymptotic behavior of solutions in an exterior domain, for example see [9,15,16,8,14,13,2].…”

“…See also [24] for a similar result for u ∈ L p (R 2 ) with p > 1 on the generalized Newtonian fluid. Other types of Liouville properties for the stationary Navier-Stokes equation on the plane were also extensively studied, such as under the growth condition lim sup |x| −α |u(x)| < ∞ as |x| → ∞ for some α > 0, see [8,1]; existence and asymptotic behavior of solutions in an exterior domain, see [12,20,21,13,19,16,3]. For more references on Liouville theorems of (3), we refer to [9,4,23,14,6] and the references therein.…”

“…ν|∇w| 2 dx + C + Cτ (τ − ρ) 2 + Cτ (τ − ρ) 4 ,provided b L 1 is small enough and we used(12) and Corollary 4.2.Finally, by settingg(r) = ˆBr |∇h| 2 + |∇w| 2 dx,we arrive atg(ρ) ≤ 1 2 g(τ ) + C + CR τ − ρ + CR (τ − ρ) 4 ,where3R 4 ≤ ρ < τ ≤ R. Hence, by Lemma 2.3, we haveˆBR/2 |∇w| 2 + |∇h| 2 dx ≤ C + C R 3 .…”

Asymptotic behavior of the steady Navier-Stokes flow in the exterior domain

“…Note that the decay is far from the prior estimates. For example, Gilbarg-Weinberger [12] showed the velocity v(x) = o(log the Dirichlet energy is bounded in an exterior domain For more references on this topic, we refer to [5,10,18] and the references therein.…”

Boundary geometry and regularity of solution to the compressible Navier-Stokes equations in bounded domains of ℝn

“…Among the references given in this paper, in bounded domains in R n one may refer to [3] for zero velocity boundary value problem, [4] for a non zero velocity boundary condition and a discontinuous density inflow boundary condition in a rectangle domain, [9] for nonzero velocity and pressure boundary condition in the unit disk, [8,10] for nonzero velocity and pressure boundary condition in polygonal domains, and [12] for initial and boundary value problem in smooth bounded domains. In exterior domains one may refer to [6] for nonzero boundary condition and [7,11] for zero velocity boundary value problem. This paper is an extension of [9] to a bounded domain contained in the unit ball of R n .…”