Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

Abstract

Abstract:In this paper we consider the system of the non-steady Navier-Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y , respectively, to be the space of "possible" solutions of this problem and the space of its data. We define the operator N : X → Y and formulate our problem in terms of operator equations. Let u ∈ X and G P u : X → Y be the Frechet derivative of N at u. We prove that G P u is one-to-one and onto Y . Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. Next result proved in the Appendix of this paper is W 2,2 -regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

“…At the outlet of a channel the boundary condition ν ∂v ∂n − pn = σ (cf. [20][21][22]52,53] and references therein) or νε(v)n − pn = 0 (cf. [54,53] and references therein) also is used.…”

The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

“…2. If u D ∈ W 1/2,2 (∂Ω D ), then there exists a function U ∈ W 1,2 (Ω) which satisfies the last condition in (7). Moreover,…”

“…where Re > 0 is the Reynolds number of the system and n is the outer normal to the boundary. At present, (DN) is a well-established boundary condition to represent natural outflows (see, e.g., [7] and the references therein). For a recent discussion on the derivation and physical meaning of the do-nothing condition (DN), we refer to [31].…”

Green Functions of Conormal Derivative Problems for Stationary Stokes System

“…(1. 2) We prove that if the Poincaré inequality (2.3) holds in Ω and weak solutions of the Stokes system with bounded data are continuous in the interior of the domain, then the Green function exists and satisfies the pointwise bound (1.2) away from the boundary of the domain; see Theorem 2.4. The Green function satisfies the pointwise bound (1.2) globally if we further assume that weak solutions of the Stokes system with bounded data are locally bounded up to the boundary; see Theorem 2.6.…”

“…The conormal derivative boundary condition arises from the variational principle, and one may consider this type of condition on the output of the channel where the velocity of the flow is a prior unknown when describing a flow through a finite channel. See [22,2] and references therein. Our focus in this paper is to find minimal regularity assumptions on the coefficients and on the boundary of the domain for the existence of the Green function (G, Π) satisfying the pointwise bound |G(x, y)| ≤ C|x − y| 2−d , x = y.…”