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Plane Kolmogorov flows and Takens-Bogdanov bifurcation without parameters: the singly reversible case. (English) Zbl 1233.37054
The authors consider the motion of a viscous incompressible plane fluid flow governed by the Navier-Stokes equations, which, in the balance-of-momentum form, is given by \[ \frac{\partial \varrho u_i}{\partial t}+\frac{\partial \Pi_{ij}}{\partial x_j}=\varrho \sigma F_i, \quad \frac{\partial u_j}{\partial x_j}=0, \] where the momentum flux density is given by \(\Pi_{ij}=\varrho u_i u_j+p\delta_{ij}-\varrho \nu(\partial_{x_j}u_i+ \partial_{x_i}u_j)\), \(\mathbf{u}=(u_1,u_2)^T\) denotes the velocity field, \(p\) is the pressure, \(\varrho\) the density, and \(\nu\) denotes the viscosity. Thus it is considered the plane Kolmogorov fluid flows in a cylindrical domain \(K=\mathbb{R}\times S^1_{2\pi}\), where \(x_1\in \mathbb{R}\), \(x_2 \in S^1_{2\pi}\), and \(S^1_{a}:=\mathbb{R}/a\mathbb{Z}\). Fluid motion is generated by the action of a body force \(\sigma\mathbf{F}\), where \[ \mathbf{F}(x_1,x_2)=(F_1(x_2),0)', \quad F_1(x_2)=F_1(x_2+2\pi). \] It is convenient to single out the basic steady state of the problem \(\widetilde{\mathbf{u}}_{\ast}=\frac{\sigma}{\nu}(U(x_2),0)\), \(p_{\ast}=0,\) where \(U(x_2)\) is the solution of the equation \(U''(x_2)+F(x_2)=0\) with vanishing mean value. The Reynolds number \(R=\sigma \nu^{-2}L\) is introduced, where \(L=1\) is the unit of length, \(\varrho=1\), and \(\sigma/\nu\) is taken as the velocity unit.
Looking for time-independent bounded solutions near the critical Reynolds number, a dynamic system on a \(6\)-dimensional center manifold is obtained, which is invariant by translations in the unbounded spatial direction. Its reduction by first integrals leads to a \(3\)-dimensional system with a line of equilibria. At isolated points the normal hyperbolicity of this line fails, stipulated by a transverse double zero eigenvalue that leads to the investigated Takens-Bogdanov bifurcation. The simpler case of a double symmetry of external force and corresponding bi-reversible system was investigated in a previous work of the authors [Asymptotic Anal. 60, No. 3–4, 185–211 (2008; Zbl 1173.35093)]. In the case of only one symmetry, as studied in this article, the complete set \(\mathcal{B}\) of equilibria and multipulse heteroclinic pairs is given.
MSC:
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
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