Abstract
The stationary Navier–Stokes equations under Navier boundary conditions are considered in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and of the strength of the external force. For some particular forcing, it is shown that uniqueness persists on some continuous branch of solutions, when these quantities become arbitrarily large. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric solutions. This proof is computer-assisted, based on a local representation of branches as analytic arcs.
Highlights
Introduction and Main ResultsLet Ω ⊂ R2 be a bounded domain and consider the stationary Navier–Stokes equations− νΔu + (u · ∇)u + ∇p = f, ∇ · u = 0 in Ω, (1)that model the steady-state motion of an incompressible viscous fluid: u is its velocity, p its pressure, f is an external force, ν > 0 is the kinematic viscosity
Non-uniqueness has been obtained in very particular situations such as the Benard problem [25], see [20] for the same problem tackled through computer assistance, or the so-called Taylor problem, where one has multiplicity of solutions if f is large, see [32] and [15, Theorem IX.2.2] for a slightly more general statement
The remaining part of the proof of Theorem 2 is obtained through computer assistance and is described in Sect. 4, where we provide the explicit definition of the function f and the value ν0 of the bifurcation point
Summary
Let Ω ⊂ R2 be a bounded domain and consider the stationary Navier–Stokes equations. that model the steady-state motion of an incompressible viscous fluid: u is its velocity, p its pressure, f is an external force, ν > 0 is the kinematic viscosity. Non-uniqueness has been obtained in very particular situations such as the Benard problem [25], see [20] for the same problem tackled through computer assistance, or the so-called Taylor problem, where one has multiplicity of solutions if f is large, see [32] and [15, Theorem IX.2.2] for a slightly more general statement. A complete comprehension of these phenomena is a challenging task, see [21, Problem 67]. In some situations, such as in 2D geophysical models [24], (2) is no longer suitable to describe the behavior of the fluid at the boundary and a slip boundary condition appears more realistic. In 1827, Navier [22] proposed boundary conditions with friction, with a stagnant layer of fluid close to the wall allowing a fluid
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