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.

In this work, we modify the weighted (L^p) bounds for elements of the Hilbert space (tilde{D}^{1,2}(Omega )). Using this bound, we derive the upper bound for the density, which is the key issue to global solution provided the shear viscosity is a positive constant and the bulk one is (lambda = rho ^{beta }) with (beta >4/3). Our results extend the earlier results due to Vaigant-Kazhikhov (Sib Math J 36:1283–1316, 1995) where they required that (beta >3), initial densities is strictly away from vacuum, and that the domain is bounded.

In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in (B_{infty ,1}^0). Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in all critical spaces (B_{p,1}^frac{1}{p}) with (pin [1,infty ]).

Asymptotic expansion in far-field for the incompressible Navier–Stokes flow are established. It is well known that a velocity decays slowly in far-field. This property prevents classical procedure giving asymptotic expansions of solutions with high-order. In this paper, under natural settings and moment conditions on the initial vorticity, technique of renormalization together with Biot–Savart law derives an asymptotic expansion for velocity with high-order. Especially scalings and large-time behaviors of the expansions are clarified. By employing them, time evolution of velocity in far-field is drawn. As an appendix, asymptotic behavior of solutions as time variable tends to infinity is given. In this assertion, large-time behavior of velocity is discovered clearly.

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic (W^{2,p})-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

The classical problem of removable singularities is considered for solutions to the stationary Navier–Stokes system in dimension (nge 3) and an old theorem of Shapiro (TAMS 187:335–363, 1974) is recovered and extended to solutions in a half ball vanishing on the flat boundary. Moreover, for (n=4) it is proved that there are not distributional solutions, smooth away from the singularity and such that (u(x)=O(|x|^{-1})).

In this paper, we establish Liouville-type theorems for the stationary ideal compressible magnetohydrodynamics system in (textbf{R}^n) with (nin {1, 2, 3}). We address various cases when the finite energy condition is in force and the stationary density function (rho ) satisfies (displaystyle lim _{|x|rightarrow infty }rho (x)=rho _infty ge 0). Our proof relies heavily on the good structure of the nonlinear magnetic force term and the usage of well-chosen smooth cut-off test functions.

In this article, we investigate the incompressible 2D Euler equations on a rotating biaxial ellipsoid, which model the dynamics of the atmosphere of a Jovian planet. We study the non-zonal Rossby–Haurwitz solutions of the Euler equations on an ellipsoid, while previous works only considered the case of a sphere. Our main results include: the existence and uniqueness of the stationary Rossby–Haurwitz solutions; the construction of the traveling-wave solutions; and the demonstration of the Lyapunov instability of both the stationary and the traveling-wave solutions.

In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of liquid, i.e. a liquid contained between two solid surfaces and part of the liquid surface is in contact with the air. The liquid is governed by Navier–Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the liquid thickness (lubrication approximation) and the contact angle between the liquid–solid and the liquid–gas interfaces close to (pi ). This resulting model is a free boundary problem for the equation (h_t + (h^3h_{xxx})_x = 0), for which we have (h>0) at the contact point (different from the usual thin film equation with (h=0) at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Furthermore, the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from the no-slip condition. Additionally, we show the global stability of steady state solutions in a periodic setting.