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In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture L 2 -convergence result via auxiliary logarithmic time decay estimates with the initial data in L p (R 2) ∩ L 2 (R 2). Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized. [ABSTRACT FROM AUTHOR]

The present paper deals with non-Newtonian viscoelastic flows of Oldroyd-B type in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier–Stokes/Oldroyd-B system towards the asymptotic model. [ABSTRACT FROM AUTHOR]

In this paper, we study the behaviour, in terms of compressibility, of steady flows of weakly compressible viscoelastic fluids having a differential constitutive law. The models considered here are Jeffreys' and Maxwell's. In both cases, we establish the existence of an asymptotic expansion in the neighbourhood of the steady incompressible fluid flow, with respect to the vanishing isothermal compressibility coefficient. [ABSTRACT FROM AUTHOR]