The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.
In this paper, we investigate the decay rate of stabilization of the solution of the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as a function of the order parametr. We prove that for appropriate constants, which appear in the above-mentioned model, we can decide upon the exponencial decrease of the solution to its attractor for time tending to infinity.
We consider a sequence of curved rods which consist of isotropic material and which are clamped on the lower base or on both bases. We study the asymptotic behaviour of the stress tensor and displacement under the assumptions of linearized elasticity when the cross-sectional diameter of the rods tends to zero and the body force is given in the particular form. The analysis covers the case of a non-smooth limit line of centroids. We show how the body force and the choice of the approximating curved rods can affect the strong convergence and the limit form of the stress tensor for the curved rods clamped on both bases.