Non-linear asymptotic stability for the through-passing flows of inviscid incompressible fluid

Abstract

The paper addresses the dynamics of inviscid incompressible fluid confined within bounded domain with the inflow and outflow of fluid through certain parts of the boundary. This system is non-conservative essentially since the fluxes of energy and vorticity through the flow boundary are not equal to zero. Therefore, the dynamics of such flows should demonstrate the generic non-conservative phenomena such as the asymptotic stability of the equilibria, the onset of instability or the excitation of the self-oscillations, etc. These phenomena are studied extensively for the flows of the viscous fluids but not for the inviscid ones. In this paper, we prove a sufficient condition for the non-linear asymptotic stability of the inviscid steady flows. In this paper, we consider the flows of inviscid incompressible fluid through a given domain thereby assuming that the fluid is allowed to come in and out the flow domain through the prescribed parts of its boundary. Such sort of boundaries and the related fluid flows are referred to as open ones. The relevant examples are the flows through the finite ducts or pipes with the inflow of fluid at one end and outflow on the other end. In the case of open boundaries, the inviscid fluid represent a non-conservative system. The violation of the conservation laws takes place since the inflow and outflow of fluid supplies to and withdraws from the flow both energy and vorticity. The withdrawal can be considered as some sort of dissipation. Therefore, the dynamics of open flows should demonstrate the generic non-conservative phenomena such as the asymptotic stability of the equilibria, the onset of the instability or excitation of the self-oscillations, etc. These features are quite new for the dynamics of inviscid fluid though they are investigated widely for the Navier-Stokes and allied equations. Generally, the result of the withdrawal-supplying competition depends on boundary conditions at the flow inlet and outlet and the withdrawal dominates sometimes. For example, Morgulis and Yudovich (11) have pointed out wide classes of 2D open flows admitting the decreasing Liapunov functionals. Their construction employs the Arnold stability approach. Later on, Gallaire and Chomaz (7) examined the effect of the boundary conditions on the inviscid swirling flows through the finite pipes. Among other