On the weak solvability of the problem of viscoelasticity with memory

Abstract

The existence of a weak solution is proved for a certain Oldroyd model of motion of a viscoelastic medium that allows for the memory of the system. The proof uses the theory of regular Lagrange flows and a topological approximation method that reduces the posed problem to an operator equation, its ε-regularization in smoother spaces, the use of a priori estimates and a topological degree for the proof of the solvability of the ε-regularized equations, and the passage to the limit as ε → 0.