On existence of a classical solution to a generalized Kelvin-Voigt model (Q1955979)

The authors consider the generalized Kelvin-Voigt model describing the motion of a two-dimensional compressible visco-elastic body \[ \rho_0 u_{tt} - \text{div}\, T = \rho_0 f \] in the time-space cylinder \(Q \subset R\times R^2\), together with the space-periodic boundary conditions. Assuming the density \(\rho_0\) time-independent and the stress tensor \(T = T_e + T_v\) with \[ T_e = T_e(D(u)) \sim (1+ |D(u)|^2)^{(q-2)/2}, q \in (1,2] \] and \[ T_v = T_v(D(u_t)) \sim (1+ |D(u_t)|^2)^{(r-2)/2}, r \in [2,\infty) \] (here, \(D(v)\) stands for the symmetric part of \(\nabla v\)), the authors show the existence of a global-in-time classical solution to this problem, provided the initial condition is smooth enough, however arbitrarily large. The proof is based on Meyers estimates giving slightly higher regularity of the solution which further implies Hölder continuity of \(\nabla u_t\).