Local Solvability of a Boundary Value Problem for One-Dimensional Motion of a Granular Matter

Abstract

This paper investigates the motion of a granular medium for a shallow, vertically shaken bed. Granular matter is one of the most common in nature, and its study has received much attention in recent decades. On the one hand, such a matter behaves like a fluid and has the ability to take the form of its container and to leak away. On the other hand, its behavior is similar to a solid. This work assumes the Leidenfrost state as an initial state with granular matter resembling a fluid heated up from below. The goal is to establish the theorem on the local solvability of the initial-boundary value problem for the one-dimensional motion of a granular medium with consideration to vibrations and scopes of the hydrodynamical approach. The introduction gives brief overviews of the problem and related studies. The Section 1 discusses the one-dimensional isothermal problem for the motion of a granular matter, which is treated like a continuous medium, within the scopes of the hydrodynamical model. The original set of equations is rearranged, and the theorem on the existence of a generalized solution is established. The Section 2 proves the local temporal solvability of the initial-boundary problem Sobolev's and Holder's spaces.