Journal of Mathematical Fluid Mechanics最新文献

The paper deals with the problem of the energy conservation for the weak solutions to the compressible Primitive Equations (CPE) system with degenerate viscosity. The sufficient conditions on the regularity of weak solutions for the energy equality are obtained even for the case when the solutions may include vacuum. In this paper, we show two theorems, the first one gives regularity in the classical isotropic Sobolev and Besov spaces. The second one states regularity in the anisotropic spaces. We obtain new regularity results in the second theorem due to the special structure of CPE system, which are in contrast to compressible Navier-Stokes equations.

The main concern of this paper is to mathematically investigate the formation of a plasma sheath near the surface of nonplanar walls. We study the existence and asymptotic stability of stationary solutions for the nonisentropic Euler-Poisson equations in a domain of which boundary is drawn by a graph, by employing a space weighted energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Because the domain is the perturbed half space, we first show the time-global solvability of the nonisentropic Euler-Poisson equations, then construct stationary solutions by using the time-global solutions.

This paper studies stationary supersonic compressible Euler flow in a two-dimensional finite straight nozzle. By introducing Glimm functionals with variable weights, we overcome the potential accumulation of successive reflections of weak waves between the two lateral walls, thus establish the existence of a weak entropy solution to a boundary-value problem of the Euler equations in the space of functions with bounded variations by a modified Glimm scheme.

We are concerned with the steady Magnetohydrodynamics(MHD)-heat system with Joule effects under mixed boundary conditions. The boundary conditions for fluid may include the stick, pressure (or total pressure), vorticity, stress (or total stress) and friction types (Tresca slip, leak, one-sided leaks) boundary conditions together and for the electromagnetic field non-homogeneous mixed boundary conditions are given. The conditions for temperature may include non-homogeneous Dirichlet, Neumann and Robin conditions together. The viscosity, magnetic permeability, electrical conductivity, thermal conductivity and specific heat of the fluid depend on the temperature. The domain for fluid is not assumed to be simply connected. For the problem involving the static pressure and stress boundary conditions for fluid it is proved that if the parameter for buoyancy effect is small in accordance with the data of problem, a datum concerned with non-homogeneous mixed boundary conditions for magnetic field and the data of problem are small enough, then there exists a solution. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proved when the parameter for buoyancy effect is small in accordance with the data of problem, a datum concerned with non-homogeneous mixed boundary conditions for magnetic field is small, but without the auxiliary smallness of the other data of problem. In addition (Appendix), a very simple proof of the fact that vorticity quadratic form for vector fields with mixed boundary conditions is positive-definite, which has been known in a previous paper and is used in this paper, is given.

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.