Abstract
We consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.
Highlights
Introduction and Main ResultWe consider the Navier–Stokes equations∂tu − νΔu + (u · ∇)u + ∇p = f, ∇ · u = 0 on Ω, (1.1)for the velocity u = u(t, x, y) of an incompressible fluid on a planar domain Ω, satisfying suitable boundary conditions for (x, y) ∈ ∂Ω and initial conditions at t = 0
Following [28], we impose Navier boundary conditions on ∂Ω, which are given by u1 = ∂xu2 = 0 on {0, π} × (0, π), u2 = ∂yu1 = 0 on (0, π) × {0, π}
Navier boundary conditions are appropriate in many physically relevant cases [3], which includes the presence of permeable walls [4] or turbulent boundary layers [13,16]
Summary
For the velocity u = u(t, x, y) of an incompressible fluid on a planar domain Ω, satisfying suitable boundary conditions for (x, y) ∈ ∂Ω and initial conditions at t = 0. Navier boundary conditions are appropriate in many physically relevant cases [3], which includes the presence of permeable walls [4] or turbulent boundary layers [13,16]. A fair amount is known about the (non)uniqueness of stationary solutions. This includes the existence of a bifurcation between curves of stationary solutions with different symmetries [28]. We prove the existence of a Hopf bifurcation for the Eq (1.1) with boundary conditions (1.2), and with a forcing function f that satisfies (∂xf2 − ∂yf1)(x, y) = 5 sin(x) sin(2y) − 13 sin(3x) sin(2y)
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