On the existence, uniqueness and regularity of solutions of a viscoelastic Stokes problem modelling salt rocks

TL;DR

This paper investigates the mathematical properties of a viscoelastic Stokes problem modeling salt rocks, establishing conditions for existence, uniqueness, and regularity of solutions in Sobolev spaces, with explicit solution estimates.

Contribution

It provides new conditions ensuring the well-posedness and regularity of solutions for a viscoelastic salt rock model, including higher-order regularity under relaxed assumptions.

Findings

Conditions for uniform ellipticity are established.

Existence and uniqueness of solutions are proved.

Explicit estimates for solutions are provided.

Abstract

A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of $L^2$-based Sobolev spaces. The system is transformed into a generalized Stokes problem, and the proper conditions on the parameters of the model that guarantee that the system is uniformly elliptic are given. Under those conditions, existence, uniqueness and low-order regularity are obtained under classical regularity conditions on the data, while higher-order regularity is proved under less stringent conditions than classical ones. Explicit estimates for the solution in terms of the data are given accordingly.