Journal of Mathematical Fluid Mechanics最新文献

In this article, the 3D Voigt-regularized Magnetohydrodynamic equations are considered, for which it is unknown if the uniqueness of weak solution exists. First, we prove that the uniform global attractor exists by constructing an evolutionary system. Then singular limits of this system are established. Namely, when a certain regularization parameter disappears, the convergence of global attractors is shown between the 3D autonomous Voigt-regularized Magnetohydrodynamic equations and Magnetohydrodynamic equations.

In this paper, we present an exact solution for the nonlinear governing equation coupled with relevant boundary conditions, which arise from the study of Saturn’s stratified circumpolar atmospheric flow. The solution is explicit in the Lagrangian framework by specifying its hypotrochoidal particle paths. An instability result of such nonlinear waves is also obtained by means of the short-wavelength instability approach.

We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov (Geom Funct Anal (GAFA) 29(1):190–197, 2019), and we study the corresponding fluid particle dynamics. This is an ode analysis, which contributes to the description of Gavrilov’s vector field.

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.