Novosibirsk, Russia, May, 30 – June, 4, 2011

"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

## Gubarev Y.G.## On the long-wave instability of stationary plane-parallel flows of a homogeneous in density ideal incompressible fluid with a free boundary in the gravity fieldThe linear stability problem of steady-state plane-parallel flows of a homogeneous in density inviscid incompressible fluid with a free surface in the gravity field is studied. It is proved by the energy method that for such flows there are no sufficient conditions for stability against small plane long-wave perturbations. However, earlier V.M. Teshukov by means of the generalized characteristics method has been received the hyperbolicity conditions for equations describing the long waves propagation on a free boundary of horizontal layer of a whirling homogeneous in density ideal incompressible fluid in the gravity field. At that, the given hyperbolicity conditions he interpreted as sufficient conditions for linear stability exactly. As presence or absence of sufficient conditions for stability shouldn't depend on choice of either research method, there is an urgent need to eliminate the arisen contradiction. With that end in view, an analytical example of the stationary plane-parallel flow and small plane long-wave perturbations in the form of normal modes imposed on it has been constructed (together with E.Yu. Knyazeva). It was found out that the given perturbations don't fall under action of Teshukov’s hyperbolicity conditions. In particular, for their time growth it is absolutely indifferent, these hyperbolicity conditions are satisfied as such or not. The reason for similar state of affairs consists that Teshukov’s hyperbolicity conditions are fair not for all possible small plane long-wave perturbations, and only for their some subclass (besides, that fundamentally, not being independent). Based on the above-stated, it seems logical that studying of the given problems should be continued, and in the following two main directions: 1) proof of absolute instability for steady-state plane-parallel flows of a homogeneous in density inviscid incompressible fluid with a free surface in the gravity field against small plane long-wave perturbations and 2) search of new hyperbolicity conditions which would be expressed through integrals of motion and, thus, would be isolated independent particular classes of long waves.
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