Existence and Uniqueness of the Initial-Boundary Value Problem Solution of Multicomponent Sediments Transport in Coastal Marine Systems

Abstract

Introduction. This work is devoted to the study of a non-stationary two-dimensional model of sediment transport in coastal marine systems. The model takes into account the complex multi-fractional composition of sediments, the gravity effect and tangential stress caused by the impact of waves, turbulent exchange, dynamically changing bottom topography, and other factors. The aim of the work was to carry out an analytical study of the conditions for the initialboundary value problem existence and uniqueness corresponding to the specified model.Materials and Methods. Linearization of the initial-boundary value problem is performed on a temporary uniform grid. The nonlinear coefficients of a quasilinear parabolic equation are taken with a “delay” by one grid step. Thus, a chain of correlated by initial conditions is the final solutions of problems is built. The study of the existence and uniqueness of the problems included in this chain, and therefore the original problem as a whole, is carried out involving the methods of mathematical and functional analysis, as well as methods for solving differential equations.Results. Earlier, the authors investigated the existence and uniqueness of the initial-boundary value problem of the transport of sediments of a single-component composition. In the present work, the result obtained is extended to the case of multi-fractional sediments.Discussion and Conclusions. The non-linear spatial two-dimensional model of sediment transport was previously investigated by the team of authors in the case of bottom sediments consisting of particles having the same characteristic dimensions and density (single-component composition) based on the analysis of the existing results of mathematical modeling of hydrodynamic processes. In this paper, the previous results of the study are extended to the case of sediments of a multicomponent composition, namely, the conditions for the existence and uniqueness of the solution of the initial-boundary value problem corresponding to the considered model are determined.