Stability and instability in adiabatic shearing motions of incompressible fluids

We investigate the effect of temperature dependence of the viscosity on the stability of the adiabatic shearing flows of an incompressible Newtonian viscous fluid between two parallel plates. When the viscosity strongly decreases with temperature, the shearing flow caused by a steady motion of the upper plate (steady shearing) becomes unstable, while the shearing flow caused by a time-dependent body force is found to be stable.     

Global existence for a system of nonlocal PDEs with applications to chemically reacting incompressible fluids

We show global existence for a class of models of fluids that change their properties depending on the concentration of a chemical. We allow that the stress tensor in (t, x) depends on the velocity and concentration at other points and times. The example we have in mind foremost are materials with memory.     

Global existence of three dimensional incompressible MHD flows

We investigate the initial value problem for the three‐dimensional incompressible magnetohydrodynamics flows. Global existence and uniqueness of flows are established in the function space , provided that the norm of the initial data is less than the minimal value of the viscosity coefficients of the flows. Copyright © 2016 John Wiley & Sons, Ltd.     

Global solutions for the incompressible rotating stably stratified fluids

We consider the initial value problem of the 3D incompressible Boussinesq equations for rotating stratified fluids. We establish the dispersive and the Strichartz estimates for the linear propagator associated with both the rotation and the stable stratification. As an application, we give a simple proof of a unique existence of global in time solutions to our system using just a contraction mapping principle.     

Global weak solutions for variable density asymmetric incompressible fluids

We establish the existence of global in time weak solutions for the equations of asymmetric incompressible fluids with variable density, when the initial density is not necessarily strictly positive.     

Global strong solutions for the incompressible micropolar fluids equations

Archiv der Mathematik - We study the Cauchy problem for the system of equations describing the nonstationary flow of micropolar incompressible fluids in $$mathbb {R}^3$$ . We show the global...     

Global existence results for Oldroyd-B fluids in exterior domains

In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough.     

Global existence for three‐dimensional incompressible isotropic elastodynamics via the incompressible limit

The existence of global‐in‐time classical solutions to the Cauchy problem for incompressible nonlinear isotropic elastodynamics for small initial displacements is proved. Solutions are constructed via approximation by slightly compressible materials. The energy for the approximate solutions remains uniformly bounded on a time scale that goes to infinity as the material approaches incompressibility. A necessary component to the long‐time existence of the approximating solution is a null or linear degeneracy condition, inherent in the isotropic case, which limits the quadratic interaction of the shear waves. The proof combines energy and decay estimates based on commuting vector fields and a compactness argument. © 2004 Wiley Periodicals, Inc.     

Global existence for three‐dimensional incompressible isotropic elastodynamics

The existence of global‐in‐time classical solutions to the Cauchy problem for incompressible, nonlinear, isotropic elastodynamics for small initial displacements is proved. The generalized energy method is used to obtain strong dispersive estimates that are needed for long‐time stability. This requires the use of weighted local decay estimates for the linearized equations, which are obtained as a special case of a new general result for certain isotropic symmetric hyperbolic systems. In addition, the pressure that arises as a Lagrange multiplier to enforce the incompressibility constraint is estimated as a nonlinear term. The incompressible elasticity equations are inherently linearly degenerate in the isotropic case; i.e., the equations satisfy a null condition necessary for global existence in three dimensions. © 2007 Wiley Periodicals, Inc.     

Global existence and large time behavior of the asymmetric fluids

In this paper, we consider the asymptotic stability of the steady state with the constant equilibrium state. Under the assumptions that the \({H^3}\) norm of the initial data is small, but its higher-order derivatives could be large, we prove the global existence to the Cauchy problem for the asymmetric fluids in \({\mathbb{R}^3}\). Moreover, we obtain the time decay rates of the solutions and their higher-order spatial derivatives by introducing the negative Sobolev and Besov spaces.     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).     

Global weak solutions for asymmetric incompressible fluids with variable density

We establish the existence of global in time weak solutions for the equations of asymmetric incompressible fluids with variable density, when the initial density is not necessarily strictly positive. To cite this article: P. Braz e Silva, E.G. Santos, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Global existence and long-time asymptotics for rotating fluids in a 3D layer

The Navier–Stokes–Coriolis system is a simple model for rotating fluids, which allows to study the influence of the Coriolis force on the dynamics of three-dimensional flows. In this paper, we consider the NSC system in an infinite three-dimensional layer delimited by two horizontal planes, with periodic boundary conditions in the vertical direction. If the angular velocity parameter is sufficiently large, depending on the initial data, we prove the existence of global, infinite-energy solutions with nonzero circulation number. We also show that these solutions converge toward two-dimensional Lamb–Oseen vortices as t→∞.     

A discrete solenoidal finite difference scheme for the numerical approximation of incompressible flows

Summary For a well known class of finite difference schemes for approximating incompressible flows it is shown that the condition of discrete incompressibility can be incorporated into the discrete space. This simplifies the structure of the linear or nonlinear discrete systems and reduces the number of unknowns.     

Global existence and exponential stability of solutions to the one-dimensional full non-Newtonian fluids

This paper is concerned with global existence and exponential stability of solutions for a class of full non-Newtonian fluids with large initial data in a bounded domain Ω?(0,1).     

On the local existence for the Euler equations with free boundary for compressible and incompressible fluids

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in $H^3$, with the $H^3$ condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.