Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω₀×(0,L)∈R³. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L_∞(0,L;L₂(Ω₀)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.     

On existence of global solutions to the Navier-Stokes equations for compressible and viscous flows on the surface of a sphere

Following tecniques proposed by A. V. Khazikhov and V. A. Weigant in 1995, we prove the global, with respect to time, existence and uniqueness of the solution to the Navier-Stokes equations for a compressible, viscous and barotropic fluid which moves on the surface of a sphere. In obtaining the main estimates we make use of the Hodge decomposition and the generalized potential theory due to K. Kodaira.

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Sunto Seguendo le tecniche proposte da A. V. Khazikhov and V. A. Weigant nel 1995, si prova l'esistenza ed unicità della soluzione per le equazioni di Navier-Stokes per un fluido comprimibile, viscoso e barotropico che si muova sulla superficie di una sfera. Nell'ottenere le principali stime si utilizzano la decomposizione di Hodge e la teoria del potenziale generalizzato, dovuta a K. Kodaira.

     

On global motion of a compressible viscous heat-conducting fluid bounded by a free surface

We consider the motion of a viscous compressible heat-conducting fluid in ³ bounded by a free surface which is under constant exterior pressure. We present the global existence theorems in two cases: when the free surface is under the surface tension and without it.     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.      

On existence and regularity of solutions for 2‐D micropolar fluid equations with periodic boundary conditions

In this paper, we prove the existence and uniqueness of a global solution for 2‐D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor. Copyright © 2006 John Wiley & Sons, Ltd.     

1-D compressible viscous micropolar fluid model with non-homogeneous boundary conditions for temperature: A local existence theorem

We consider non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are assumed. Using the Faedo-Galerkin method we prove a local-in-time existence of a generalized solution.     

Global existence versus blow‐up results for one dimensional compressible Navier–Stokes equations with Maxwell's law

We consider one dimensional isentropic compressible Navier–Stokes equations with constitutive relation of Maxwell's law instead of Newtonion law. For this new model, we show that for small initial data, a unique smooth solution exists globally and converges to the equilibrium state as time goes to infinity. For some large data, in contrast to the situation for classical compressible Navier–Stokes equations, which admits global solutions, we show finite time blow up of solutions for the relaxed system. Moreover, we prove the compatibility of the two systems in the sense that, for vanishing relaxation parameters, the solutions to the relaxed system are shown to converge to the solutions of classical system.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous,incompressible fluids

We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.     

Global existence of solutions for one-dimensional compressible navier-stokes equations in the half space

We prove the existence of global solutions to the initial-boundary-value problem on the half space R＋ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.     

Stability of two‐dimensional magnetohydrodynamic motions in the periodic case

We investigate the stability of two‐dimensional periodic solutions to magnetohydrodynamics equations in the class of three‐dimensional periodic solutions. We show the existence of global, strong three‐dimensional solutions to magnetohydrodynamics equations, which are close to two‐dimensional solutions. The advantage of our approach is that neither these solutions nor the external forces have to vanish at infinity. Copyright © 2015 John Wiley & Sons, Ltd.     

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball,III: The 3-D Boltzmann equation

This paper is a continuation of the works in [35] and [37], where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function.     

On a non‐stationary fluid flow problem in an infinite periodic pipe

We study linearized, non‐stationary Navier–Stokes type equations with the given flux in an infinite pipe periodic of period length L with respect to . The existence and uniqueness of the solution is proved. Moreover, the convergence of the solution in a finite pipe of length to the L‐periodic solution as is investigated.     

An improved blow‐up criterion for smooth solutions of the two‐dimensional MHD equations

Our main object is to establish a regularity criterion with p≥q > 1 for the incompressible magnetohydrodynamics equations with zero magnetic diffusivity. Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence of solutions to a magnetohydrodynamic‐omega model

In this paper, some sufficient conditions in terms of the magnetic field are established to guarantee global existence of solutions to a magnetohydrodynamic‐omega model.