Energy-based stability estimates for incompressible media with tensor-nonlinear constitutive relations

Tensor-nonlinear media are generalized models of continua having nonlinear relationship between the stress and strain rate, important for the descriptions of, for example, non-Newtonian fluids. We consider a wide class of such tensor-nonlinear, isotropic, incompressible media, which may possess a scalar stress potential in terms of velocity gradients, i.e., the analogue or Raleigh function. We also provide a setting of several linearized problems for these media in moving three-dimensional domains. Energy-based estimates and subsequent Lyapunov, asymptotic and exponential stability results are derived through an application of a sequence of integral inequalities. We also present particular cases of stability estimates for a variety of tensor-linear, or quasilinear, media that may or may not possess the dissipation potential, such as the Bingham and Saint-Venant media, or Newtonian viscous fluid.