Boundary Layer Problems in the Viscosity-Diffusion Vanishing Limits for the Incompressible MHD Systems

In this paper, we we study boundary layer problems for the incompressible MHD systems in the presence of physical boundaries with the standard Dirichlet oundary conditions with small generic viscosity and diffusion coefficients. We identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl's type boundary layers and prove rigorously that the solutions to the viscous and diffusive incompressible MHD systems converges strongly to the superposition of the solution to the ideal MHD systems with a Prandtl's type boundary layer corrector. One of the main difficulties is to deal with the effect of the difference between viscosity and diffusion coefficients and to control the singular boundary layers resulting from the Dirichlet boundary conditions for both the viscosity and the magnetic fields. One key derivation here is that for the class of initial data we identify here, there exist cancelations between the boundary layers of the velocity field and that of the magnetic fields so that one can use an elaborate energy method to take advantage this special structure. In addition, in the case of fixed positive viscosity, we also establish the stability of diffusive boundary layer for the magnetic field and convergence of solutions in the limit of zero magnetic diffusion for general initial data.Comment: This paper is translated by published paper in Chinese in "Sciences in China:Mathematics, Vol 47(2017), No.10, pp1-2

Incompressible inviscid resistive MHD surface waves in 2D

We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the lower boundary is a rigid flat bottom. We prove the global well-posedness of the inviscid and resistive problem with surface tension around a non-horizontal uniform magnetic field; moreover, the solution decays to the equilibrium almost exponentially. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field.Comment: 36p

Global Smooth Supersonic Flows in Infinite Expanding Nozzles

This paper concerns smooth supersonic flows with Lipschitz continuous speed in two-dimensional infinite expanding nozzles, which are governed by a quasilinear hyperbolic equation being singular at the sonic and vacuum state. The flow satisfies the slip condition on the walls and the flow velocity is prescribed at the inlet. First, it is proved that if the incoming flow is away from the sonic and vacuum state and its streamlines are rarefactive at the inlet, then a flow in a straight nozzle never approaches the sonic and vacuum state in any bounded region. Furthermore, a sufficient and necessary condition of the incoming flow at the inlet is derived for the existence of a global smooth supersonic flow in a straight nozzle. Then, it is shown that for each incoming flow satisfying this condition, there exists uniquely a global smooth supersonic flow in a symmetric nozzle with convex upper wall. It is noted that such a flow may contain a vacuum. If there is a vacuum for a global smooth transonic flow in a symmetric nozzle with convex upper wall, it is proved that for the symmetric upper part of the flow, the first vacuum point along the symmetric axis must be located at the upper wall and the set of vacuum points is the closed domain bounded by the tangent half-line of the upper wall at this point to downstream and the upper wall after this point. Moreover, the flow speed is globally Lipschitz continuous in the nozzle, and on the boundary between the gas and the vacuum, the flow velocity is along this boundary and the normal derivatives of the flow speed and the square of the sound speed both are zero. As an immediate consequence, the local smooth transonic flow obtained in [10] can be extended into a global smooth transonic flow in a symmetric nozzle whose upper wall after the local flow is convex

On an Elliptic Free Boundary Problem and Subsonic Jet Flows for a Given Surrounding Pressure

This paper concerns compressible subsonic jet flows for a given surrounding pressure from a two-dimensional finitely long convergent nozzle with straight solid wall, which are governed by a free boundary problem for a quasilinear elliptic equation. For a given surrounding pressure and a given incoming mass flux, we seek a subsonic jet flow with the given incoming mass flux such that the flow velocity at the inlet is along the normal direction, the flow satisfies the slip condition at the wall, and the pressure of the flow at the free boundary coincides with the given surrounding pressure. In general, the free boundary contains two parts: one is the particle path connected with the wall and the other is a level set of the velocity potential. We identify a suitable space of flows in terms of the minimal speed and the maximal velocity potential difference for the well-posedness of the problem. It is shown that there is an optimal interval such that there exists a unique subsonic jet flow in the space iff the length of the nozzle belongs to this interval. Furthermore, the optimal regularity and other properties of the flows are shown.Comment: accepted on SIAM J. Math. Ana

Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in 3-dimensional domains

In this paper, we investigate the uniform regularity for the isentropic compressible Navier-Stokes system with general Navier-slip boundary conditions (1.6) and the inviscid limit to the compressible Euler system. It is shown that there exists a unique strong solution of the compressible Navier-Stokes equations with general Navier-slip boundary conditions in an interval of time which is uniform in the vanishing viscosity limit. The solution is uniformly bounded in a conormal Sobolev space and is uniform bounded in $W^{1,\infty}$. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. In particular, it is proved that the velocity will be uniform bounded in $L^\infty(0,T;H^2)$ when the boundary is flat and the Navier-Stokes system is supplemented with the special boundary condition (1.21). Based on such uniform estimates, we prove the convergence of the viscous solutions to the inviscid ones in $L^\infty(0,T;L^2)$, $L^\infty(0,T;H^1)$ and $L^\infty([0,T]\times\Omega)$ with a rate of convergence.Comment: 54 page

Stability of Rarefaction Waves to the 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum state. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.Comment: 30 page

Global classical solutions to the two-dimensional compressible Navier-Stokes equations in R2\mathbb{R}^2

In this paper, we prove the global well-posedness of the classical solution to the 2D Cauchy problem of the compressible Navier-Stokes equations with arbitrarily large initial data when the shear viscosity $\mu$ is a positive constant and the bulk viscosity \l(\r)=\r^\b with \b>\frac43. Here the initial density keeps a non-vacuum states $\bar\rho>0$ at far fields and our results generalize the ones by Vaigant-Kazhikhov [41] for the periodic problem and by Jiu-Wang-Xin [26] and Huang-Li [8] for the Cauchy problem with vacuum states $\bar\rho=0$ at far fields. It shows that the solution will not develop the vacuum states in any finite time provided the initial density is uniformly away from vacuum. And the results also hold true when the initial data contains vacuum states in a subset of $\mathbb{R}^2$ and the natural compatibility conditions are satisfied. Some new weighted estimates are obtained to establish the upper bound of the density.Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:1207.5874, arXiv:1202.138

Uniform regularity for the free surface compressible Navier-Stokes equations with or without surface tension

In this paper, we investigate the uniform regularity of solutions to the 3-dimensional isentropic compressible Navier-Stokes system with free surfaces and study the corresponding asymptotic limits of such solutions to that of the compressible Euler system for vanishing viscosity and surface tension. It is shown that there exists an unique strong solution to the free boundary problem for the compressible Navier-Stokes system in a finite time interval which is independent of the viscosity and the surface tension. The solution is uniform bounded both in $W^{1,\infty}$ and a conormal Sobolev space. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. Based on such uniform estimates, the asymptotic limits to the free boundary problem for the ideal compressible Euler system with or without surface tension as both the viscosity and the surface tension tend to zero, are established by a strong convergence argument.Comment: 66pages. arXiv admin note: text overlap with arXiv:1202.0657 by other author

Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum

In this paper, we study the global well-posedness of the 2D compressible Navier-Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity $\mu$ is a positive constant and the bulk viscosity \l is the power function of the density, that is, \l(\r)=\r^\b with \b>3, then the 2D compressible Navier-Stokes equations with the periodic boundary conditions on the torus $\mathbb{T}^2$ admit a unique global classical solution $(\r,u)$ which may contain vacuums in an open set of $\mathbb{T}^2$. Note that the initial data can be arbitrarily large to contain vacuum states.Comment: 42 page

Remarks on Blow-up of Smooth Solutions to the Compressible Fluid with Constant and Degenerate Viscosities

In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplified and unified proof on the blow-up results to the classical solutions of the full compressible Navier-Stokes equations without heat conduction by Xin \cite{Xin} and with heat conduction by Cho-Bin \cite{CJ} will be given