On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions

Abstract

We study the nonhomogeneous boundary value problem for Navier-Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional bounded multiply connected domain Ω = Ω 1 \ Ω 2 , Ω 2 ⊂ Ω 1 . We prove that this problem has a solution if the flux F of the boundary datum through ∂Ω 2 is nonnegative (

The flux problem for the Navier-Stokes equations

“…Since w = 0 on ∂Ω, we can use (2.17) to prove that the pressure p takes constant values p j on the connected components Γ j of ∂Ω. The next result was established in [33] (Lemma 4) and [17] (Theorem 2.2) (see also [19], Remark 3.2). Taking the inner product of the Euler system (2.17) with A, we integrate the result by parts.…”

“…Thus, Bernoulli's law for solutions in Sobolev spaces must be formulated 'modulo' a negligible 'bad' subset A w with zero Hausdorff H 1 -measure. Such a version of Bernoulli's law was established in [18], Theorem 1 (see also [19], Theorem 3.2, where more details of the proof were given).…”

“…However, his method can easily be modified to prove uniform convergence on almost all level sets of any C 1 -smooth function with non-vanishing gradient. Such a modification was carried out in the proof of Lemma 3.3 in [19].…”

“…However, one can attempt to prove the required a priori bound by contradiction. This idea was first proposed by Leray [1] and has been subsequently used and modified by many authors (see, for instance, [2], [31]- [33], [17], [19]). …”

“…A novel feature in the application of this argument is the use of Bernoulli's law for Sobolev solutions of the Euler equations, which was obtained in [18] (a proof with all details was presented in [19]). The results on Bernoulli's law are based on the recent version of the Morse-Sard theorem established by Bourgain, Korobkov, and Kristensen in the joint paper [25].…”