Journal of Mathematical Fluid Mechanics最新文献

This paper aims at justifying the incompressible Navier–Stokes–Fourier limit of the steady Boltzmann equation with linear boundary condition in an exterior domain. This generalizes the work Esposito, R., Guo, Y., Marra, R.: Hydrodynamic limit of a kinetic gas flow past an obstacle. Comm. Math. Phys. 364, 765–823 (2018), to the non-isentropic case, in addition with a small external force and a small temperature variation between the wall and infinity. Some new estimates and a refined positivity-preserving scheme are established to construct a unique positive solution to the steady Boltzmann equation. An error estimate is also provided for the small Knudsen number.

We define a dissipative measure-valued (DMV) solution to the system of equations governing the motion of a general compressible, viscous, electrically and heat conducting fluid driven by non-conservative boundary conditions. We show the stability of strong solutions to the full compressible magnetohydrodynamic system in a large class of these DMV solutions. In other words, we prove a DMV-strong uniqueness principle: a DMV solution coincides with the strong solution emanating from the same initial data as long as the latter exists.

We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in (mathbb {R}^{d}) ((dge 1)), where the interaction force is given by (nabla (-Delta )^{(alpha -d)/2}rho ) with (d-2<alpha <d). It is observed by the eigenvalue analysis that the density exhibits fractional heat diffusion behavior at low frequencies, which enables us to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical (L^p) framework. Precisely, the density and its (sigma )-order derivative converge to the equilibrium at the (L^p)-rate ((1+t)^{-(sigma -sigma _1)/(alpha -d+2)}) with (-d/p-1le sigma _1< d/p-1), consistent with the rate of solutions for the frictional heat equation. A non-local hypercoercivity argument and the effective unknown (z=u+nabla Lambda ^{alpha -d}rho ) associated with the Darcy law are introduced to overcome the difficulty from the absence of hyperbolic symmetrization for first-order dissipative systems.

Three-dimensional hydrodynamic equations of ideal incompressible fluid in Lagrangian form are considered. Their explicit solution is obtained. The trajectories of fluid particles are complex spatial curves depending on four frequencies. The vortex lines precess around the vertical axis. Their shape is determined by an arbitrary function depending on the axial Lagrangian coordinate. It is shown that the rotation axis is directed to the plane of vortex lines at some nonzero angle.

This paper studies the Stokes resolvent system (-Delta textbf{u}+lambda textbf{u}+nabla rho =textbf{f}), (nabla cdot textbf{u}=chi ) in (Omega ) with the Navier condition (textbf{u}_textbf{n}=textbf{g}_textbf{n}), ([partial textbf{u}/partial textbf{n}-rho textbf{n}+btextbf{u}]_tau =textbf{h}_tau ) on (partial Omega ). Here (Omega subset {{mathbb {R}}}^2) is a bounded domain with Lipschitz boundary. (Omega ) might have holes. First we define and study weak solutions in (W^{1,2}(Omega ;{{mathbb {C}}}^2)times L^2(Omega ;{{mathbb {C}}})). Using this result we are able to prove the existence of strong solutions of the problem in Sobolev spaces (W^{s,q}(Omega ;{{mathbb {C}}}^2)times W^{s-1,q}(Omega ;{{mathbb {C}}})), in Besov spaces (B_s^{q,r}(Omega ,{{mathbb {C}}}^2)times B_{s-1}^{q,r}(Omega ;{{mathbb {C}}})) and classical solutions in the spaces ({{mathcal {C}}}^{k,alpha } ({overline{Omega }} ;{{mathbb {C}}}^2)times {{mathcal {C}}}^{k-1,alpha }({overline{Omega }} ;{{mathbb {C}}})).

This paper is devoted to the study of the Dirichlet problem for the compressible magnetohydrodynamic system with density-dependent viscosities (mu =const>0,lambda =rho ^beta ) which was first introduced by Vaigant-Kazhikhov [18] in 1995. By assuming the endpoint case (beta =1) in the radially spherical symmetric setting, we establish the global existence to strong solution of the two-dimensional system for any large initial data. This also improves the previous work of Huang-Yan [10] where they proved the similar result for (beta >1). Our main idea is to utilize the geometric structure of a 2D spherically symmetric disc and the Sobolev critical embedding inequality of spherically symmetric functions in 2D domains, as well as a refined estimate of the upper bound of the density.

In this paper, we establish the existence of the global weak admissible solution for the Cauchy problem of a N-peakon system in the sense of (H^1(mathbb {R})) space under a sign condition. Second, we claim that the global weak admissible solution for the system with the same initial data is not unique by giving a example. Finally, an image of the solutions of the above example which does not satisfy the uniqueness is given, which makes it easier to see the properties of non-uniqueness more intuitively.

This paper is concerned with a three-dimensional Keller–Segel–Navier–Stokes system incorporating singular flux limitation and superlinear production. The primary goal is to establish global existence of weak solutions under conditions ensuring that flux limitations suppress the blow-up tendencies induced by superlinear growth. More precisely, this paper focuses on the system

in a bounded domain (Omega subset mathbb {R}^3) with smooth boundary, where (0< alpha < 1) and (beta ge 1). Under the assumption (alpha > 1 - frac{1}{3beta -1}), we prove global existence of weak solutions to the Neumann problem for ((*)). This study extends the previous work by Winkler [27], in which the corresponding system with the regular sensitivity ((|nabla c|^2+1)^{-frac{alpha }{2}}) and the linear production ((beta =1)) was considered, and highlights how strong flux limitation can control the effects of superlinear growth.

We consider the Cauchy problem to the three-dimensional isentropic compressible Magnetohydrodynamics (MHD) system with density-dependent viscosities. When the initial density is linearly equivalent to a large constant state, we prove that strong solutions exist globally in time, and there is no restriction on the size of the initial velocity and initial magnetic field. As far as we know, this is the first result on the global well-posedness of density-dependent viscosities with large initial data for 3D compressible MHD equations.

In the current state of the art regarding the Navier–Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins’ inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a (C^1) velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Comm. Pure Appl. Math 72(7), 1351–1385 (2019), we conclude the uniqueness of the solution.