Journal of Mathematical Fluid Mechanics最新文献

In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations around a shear flow. Without any structural assumption, it is well-known that the optimal regularity of Prandtl is given by the class Gevrey 2 along the horizontal direction. The goal of this paper is to overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in a strip domain. We prove that the local well-posedness around a general shear flow (U_{textrm{sh}}in W^{3, infty }(0,1)) holds true, with solutions that are Gevrey class 3 in the horizontal direction.

The first order linear, fully decoupled rotational velocity projection scheme for settling 2D/3D incompressible magneto-hydrodynamic system is considered in this paper. The considered governing model is a strong nonlinear system and also a double saddle points system. The proposed scheme mainly apply the first order Euler semi implicit scheme for temporal discretization, delicate implicit–explicit treatments for handling the strong nonlinear terms, and the mixed finite element method is used for spatial discretization. Then the system can be transformed into a series of linear elliptic equations such that the all variables are fully decoupled. More importantly, the existence of rotational term in the proposed algorithm makes the theoretical analysis quite difficult to carry out. Therefore, with the help of a Gauge–Uzawa form that we derive the unconditional energy stability. The results of 2D/3D numerical simulations are proved compact with the theoretical analysis.

In this paper, we are interested in the global well-posedness of the strong solutions to the Cauchy problem on the compressible magnetohydrodynamics system with Hall effect. Moreover, we establish the convergence rates of the above solutions trending towards the constant equilibrium (({bar{rho }},0,bar{textbf{B}})), provided that the initial perturbation belonging to (H^3({mathbb {R}}^3) cap B_{2, infty }^{-s}({mathbb {R}}^3)) for (s in (0,frac{3}{2}]) is sufficiently small.

We investigate global existence and optimal decay rates of a generic non-conservative compressible two–fluid model with general constant viscosities and capillary coefficients, and our main purpose is three–fold: First, for any integer (ell ge 3), we show that the densities and velocities converge to their corresponding equilibrium states at the (L^2) rate ((1+t)^{-frac{3}{4}}), and the k((in [1, ell ]))–order spatial derivatives of them converge to zero at the (L^2) rate ((1+t)^{-frac{3}{4}-frac{k}{2}}), which are the same as ones of the compressible Navier–Stokes–Korteweg system. This can be regarded as non-straightforward generalization from the compressible Navier–Stokes–Korteweg system to the two–fluid model. Compared to the compressible Navier–Stokes–Korteweg system, many new mathematical challenges occur since the corresponding model is non-conservative, and its nonlinear structure is very terrible, and the corresponding linear system cannot be diagonalizable. One of key observations here is that to tackle with the strongly coupling terms, we will introduce the linear combination of the fraction densities ((beta ^+alpha ^+rho ^++beta ^-alpha ^-rho ^-)), and explore its good regularity, which is particularly better than ones of two fraction densities ((alpha ^pm rho ^pm )) themselves. Second, the linear combination of the fraction densities ((beta ^+alpha ^+rho ^++beta ^-alpha ^-rho ^-)) converges to its corresponding equilibrium state at the (L^2) rate ((1+t)^{-frac{3}{4}}), and its k((in [1, ell ]))–order spatial derivative converges to zero at the (L^2) rate ((1+t)^{-frac{3}{4}-frac{k}{2}}), but the fraction densities ((alpha ^pm rho ^pm )) themselves converge to their corresponding equilibrium states at the (L^2) rate ((1+t)^{-frac{1}{4}}), and the k((in [1, ell ]))–order spatial derivatives of them converge to zero at the (L^2) rate ((1+t)^{-frac{1}{4}-frac{k}{2}}), which are slower than ones of their linear combination ((beta ^+alpha ^+rho ^++beta ^-alpha ^-rho ^-)) and the densities. We think that this phenomenon should owe to the special structure of the system. Finally, for well–chosen initial data, we also prove the lower bounds on the decay rates, which are the same as those of the upper decay rates. Therefore, these decay rates are optimal for the compressible two–fluid model.

Motivated by Gilbarg–Weinberger’s early work on asymptotic properties of steady plane solutions of the Navier–Stokes equations on a neighborhood of infinity (Gilbarg andWeinberger in Ann Scuola Norm Super Pisa Cl Sci 5(2):381–404, 1978), we investigate asymptotic properties of steady plane solutions of this system on any cone-like domain of (Omega _0={(r,theta ); r>r_0, theta in (0,theta _0)} ) with finite Dirichlet integral and Navier-slip boundary conditions. It is proved that the velocity of the solution grows more slowly than (sqrt{log r}) as in Gilbarg andWeinberger (Ann Scuola Norm Super Pisa Cl Sci 5(2):381–404, 1978), while the mean value of the velocity converges to zero except the case of (theta _0=pi ). Noting that Cauchy integral formula representation does not work in these domains due to the boundary obstacle, we explore some new technical lemmas to deal with these general cases. Moreover, Liouville type theorem on these domains and the decay estimates of the pressure or the vorticity are also obtained.

We investigate the long-time behaviour of a coupled PDE–ODE system that describes the motion of a rigid body of arbitrary shape moving in a viscous incompressible fluid. We assume that the system formed by the rigid body and the fluid fills the entire space (mathbb {R}^{3}.) We extend in this way our previous results which were limited to the case when the rigid body was a ball. More precisely, we show that, under appropriate assumptions (in particular smallness ones) on the initial velocity field, the position of the rigid body converges to some final configuration as time goes to infinity. Finally, we show that our methodology can be applied in the case of several rigid bodies of arbitrary shapes moving in a viscous incompressible fluid.

Axially symmetric solutions to the Navier–Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and the angular components of the velocity and vorticity are assumed to vanish. If the norm of the initial swirl is sufficiently small, then the regularity of axially symmetric, weak solutions is shown. The key tool is a new estimate for the stream function in certain weighted Sobolev spaces.

Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of incompressible fluids is nowadays available not only for Navier–Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is nonlinear. The majority of such studies however concerns models where such a dependence is explicit (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of implicitly constituted incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick–slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.

This paper studies the structural implications of constant vorticity for steady three-dimensional internal water waves in a channel. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field is pointing in one horizontal direction while the interface is an arbitrary function of the other horizontal variable. We prove the following rigidity result: every three-dimensional traveling internal wave with bounded velocity for which the vorticities in the upper and lower layers are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional. The proof is accomplished using an entirely novel but largely elementary argument that draws connection to the problem of uniquely reconstructing a two-dimensional velocity field from the pressure.