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We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases.

In this paper, logarithmically improved regularity criteria for the Navier–Stokes and the MHD equations are established in terms of both the vorticity field and the pressure.

We study pointwise asymptotic stability of steady incompressible viscous fluids. The region of the motion is bounded. Our results of stability are based on the maximum modulus theorem that we prove for solutions of the Navier–Stokes equations. The asymptotic stability is based on a variational formulation. Since the region of the motion is bounded, the time decay is of exponential type. Of course suitable assumptions are made about the smallness of the size of the uniform norm of the perturbations at the initial data. With no restrictions, we are able only to prove an existence theorem of the perturbation local in time.

The current paper is devoted to the investigation of large-time behavior of the compressible Navier–Stokes–Fourier system in the whole space. Under the condition that $$\Vert \rho \Vert _{C^\alpha } $$ and $$\Vert \rho , T\Vert _{L^\infty }$$ possess uniform in time bound, we prove that the regular solutions converge to equilibrium with the optimal decay rate.

We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incompressible Navier–Stokes equations with initial data in weighted $$L^2$$ spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to $$L^2 ( (1+ r^2)^{-\frac{\gamma }{2}} dx ) $$ , with $$r= \sqrt{x_1^2 + x_2^2}$$ and $$\gamma \in (0, 2) $$ .

A new Generalized Phan–Thien–Tanner (GPTT) model is derived from a Lodge–Yamamoto type of network theory for the polymeric fluids. The GPTT model is developed to describe the rheological behavior of the viscoelastic fluids. In this paper, we investigate the initial-value problem for the GPTT model. Under the norm of the initial data is a small perturbation around some particular solution, we prove that the strong solution exists globally in critical Besov spaces.

The results presented in this paper are generalizations of earlier work on the linear stability of non-rotating round gas balls in equilibrium, with respect to perturbations with zero angular momentum. Here we allow a more general barotropic equation of state for the gas, a non-zero angular momentum of the equilibrium state, and we are considering arbitrary numbers of gas balls, intending to use the result later to prove non-linear stability. The result requires an energy stability condition, which we verify for a single, slowly rotating gas ball, and the restricted class of equations of state used in earlier papers.

Motivated by an equation arising in magnetohydrodynamics, we prove that Hölder continuous weak solutions of a nonlinear parabolic equation with singular drift velocity are classical solutions. The result is proved using the space–time Besov spaces introduced by Chemin and Lerner (J Differ Equ 121(2):314–328, 1995), combined with energy estimates, without any minimality assumption on the Hölder exponent of the weak solutions.

We give a refined regularity criterion for solutions of the three-dimensional Navier–Stokes equations with fractional dissipative term $$(-\Delta )^{\alpha /2}v$$ . The criterion is composed by the direction field of the vorticity and its magnitude simultaneously. Our result is a generalized of previous results by Beirão da Veiga and Berselli (Differ Integral Equ 15(3):345–356, 2002), and Zhou (ANZIAM J 46(3):309–316, 2005, Monatsh Math 144(3):251–257, 2005). Moreover, our result mentioned about the relation between the solution of the Navier–Stokes equations and the Euler equations.