The Showalter-Sidorov and Cauchy problems for the linear Dzekzer equation with Wentzell and Robin boundary conditions in a bounded domain

G. A. Sviridyuk   ; N. S. Goncharov   ; S. A. Zagrebina

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 1, pp. 50-63

Résumé

Deterministic and stochastic initial boundary value problems for the Dzekzer equation describing the evolution of the free surface of a filtering fluid in a bounded region with a smooth boundary are considered. Wentzell and Robin conditions are set on the boundary of the domain, and either the Showalter-Sidorov condition or the Cauchy condition is taken as the initial condition. Note that for the filtration model under study, the Wentzell condition is considered, which is not a classical condition. In recent years, the boundary condition has been considered in the mathematical literature from two points of view (classical and neoclassical). Since Cauchy and Showalter-Sidorov initial conditions have been studied earlier in various situations, in this work, in the particular case of classical Wentzell and Robin conditions, by methods of the theory of degenerate holomorphic semigroups, exact solutions have been constructed, which allow to determine quantitative predictions of changes in geochemical regime of groundwater under unpressurized filtration. Nelson-Glicklich derivative theory was used in stochastic case. In particular, the investigation of the set problems in the context of Wentzell boundary conditions allowed to determine the processes occurring at the boundary of two media (in the region and at its boundary).

Détail

Mots-clés : Dzekzer equation, Cauchy condition.
Keywords: deterministic and stochastic Sobolev-type equations, Nelson-Gliklikh derivative, Wentzell condition, Showalter-Sidorov condition

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