On the existence of steady-state solutions to the Navier-Stokes system for large fluxes

Abstract

In this paper we deal with the stationary Navier-Stokes problem in a domain with compact Lipschitz boundary ∂ and datum a in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of ∂ , with possible countable exceptional set, provided a is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for bounded.

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

“…An interesting contribution to the Navier-Stokes problem is due to H.Fujita and H. Morimoto [9] (see also [29]). They studied problem (1) in a domain Ω with two components of the boundary Γ 1 and Γ 2 .…”

Existence of axially symmetric weak solutions to steady MHD with nonhomogeneous boundary conditions

“…However, the problem that whether (1.5), (1.3) admit a solution or not is open for long times and usually referred as Leray's problem in literatures. For sufficiently small fluxes F j , one can also obtain the existence of weak solutions [2,6,7,9,10,12,18,25]. The existence was also known with certain symmetric restrictions on the domain and the boundary data and the forcing term (see [1,8,14,22,23,24]).…”

“…The first one reduced the nonhomogeneous case to homogeneous case by using the solenoidal extension of boundary value a into Ω, which was successively completed and clarified in [6,11,20]). The second one is based on a clever contradiction argument, which was used in [1,2,12,25]. However, the problem that whether (1.5), (1.3) admit a solution or not is open for long times and usually referred as Leray's problem in literatures.…”

A weighted setting for the stationary Navier Stokes equations under singular forcing

“…Let us first show, via a contraction argument, that provided the problem data is sufficiently small we have existence and uniqueness of solutions. Our contraction argument is rather standard, see for instance [11,Theorem 3.1] and [12,Theorem 5.6]. The main novelty in our approach seems to be the fact that, by restricting the weight to A 2 (Ω), we allow the domain to be merely Lipschitz.…”