Instability and gradual loss of smoothness inherent in flows of perfect fluid

Abstract

In this study, we investigated a specific instability inherent in flows of perfect incompressible fluid and appearing as an unlimited growth of strains in liquid particles. As a result, in three-dimensional flows, a vortex grows indefinitely with time, while for two-dimensional flows the growth of the vortex gradient is typical. In the general case, these statements remain to be hypothetical; however, their validity is beyond any doubt considering a set of examples available for particular flows and classes of flows. The problem of global unambiguous solvability for the Euler equations of perfect incompressible fluid remains to be the most urgent in the mathematical hydrodynamics. Because the local unambiguous solvability was established as early as 1920th in classical studies by N. Gunter and L. Liechtenstein, the problem reduces to deriving reasonably strong prior estimates for solutions and, above all, for a vortex. However, the satisfactory results are now known only in the twodimensional case [1‐7]. For example, if the initial data and the boundary of the flow domain are C × -smooth, it is possible to guarantee the existence of a unique C × smooth solution for all values of the time variable t . An interesting review of results and unresolved problems can be found in [8]. From the physical point of view and also from the point of view of a computer experiment, the conclusion about the smoothness of two-dimensional flows is of somewhat formal character. The examples presented in [9] have shown that the vortex gradient can grow indefinitely with time in two-dimensional flows, while in three-dimensional flows the vortex itself can grow indefinitely. Meanwhile, this is the smoothness, that is conserved uniformly at infinite intervals of time, could be considered as real and observable in natural and computer experiments. Such a conservation of the smoothness can be provided only by the global prior estimates of derivatives, which are based on the conservation laws. In the dynamics of the perfect incompressible fluid, the law of conservation of circulation (the Helmholtz‐Thomson theorem) acts also along with the law of conservation of energy. In the two-dimensional case, the vortex conservation follows from the law of conservation of circulation, but constraints are not imposed on its derivatives. In the three-dimensional case, there are no obstacles for the indefinite growth of the vortex and the deformation rate. There are likely no laws that should require the growth of a vortex for all or at least “almost all” threedimensional flows, excluding, perhaps, steady, periodic, and quasi-periodic flows. The same can be said also with respect to the vortex gradient in the twodimensional case. Therefore, such a growth, resulting in a gradual loss of the flow smoothness, sometimes takes place, and sometimes does not. At the same time, it can be assumed with confidence that the class of flows for which this phenomenon occurs is so wide that a dense set of initial data corresponds to this class everywhere in the phase space of the system. As a result, it turns out that almost any steady flow is unstable in the sense of Lyapunov in the vortex metric (max | curl v | + minor norm) in the three-dimensional case and in the vortex-gradient metric ( max |∇ω| + minor norm) in the two-dimensional case. This smoothness-loss phenomenon and a sort of the stochastic behavior associated with it were, for the first time, pointed to in [9].