Weak–strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows (original) (raw)
Related papers
2018
In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power (ρ^δ with 0<δ<1), based on some analysis of the nonlinear structure of this system, we identify the class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier [3], Jiu-Wang-Xin [11] and so on. Moreover, in contrast to the classical theory in the case of the constant viscosity, we show that one can not obtain any global regular solution whose L^∞ norm of u decays to zero as time t goes to infinity.
On Classical Solutions of the Compressible Magnetohydrodynamic Equations with Vacuum
SIAM Journal on Mathematical Analysis
In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set or decay at infinity. Moreover, we prove that the L ∞ norm of the deformation tensor of velocity gradients controls the possible blow-up (see [16][22]) for classical (or strong) solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the deformation tensor as the critical time approaches. Our criterion (see (1.12)) is the same as Ponce's criterion for 3-D incompressible Euler equations [15] and Huang-Li-Xin's criterion for the 3-D compressible Navier-stokes equations [9].
Nonlinearity
We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov (1987)], we obtain the global existence and uniqueness of strong solutions which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the motion of the flow has large oscillations and the interaction between the hydrodynamic and magnetodynamic effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant viscosity and heat conductivity case to that of nonlinear viscosity and degenerate heat-conductivity.
Blow-up criterion for the 333D non-resistive compressible Magnetohydrodynamic equations
arXiv (Cornell University), 2014
In this paper, we prove a blow-up criterion in terms of the magnetic field H and the mass density ρ for the strong solutions to the 3D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the upper bounds of (H, ρ) control the possible blow-up (see [26][32][36]) for strong solutions.
4 Blow-Up Criterion for the 3D Non-Resistive Compressible Magnetohydrodynamic Equations
2016
In this paper, we prove a blow-up criterion in terms of the magnetic field H and the mass density ρ for the strong solutions to the 3D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the upper bounds of (H, ρ) control the possible blow-up (see [26][32][36]) for strong solutions.
Weak solutions of the compressible isentropic Navier-Stokes equations
Applied Mathematics Letters, 1999
This note is devoted to some properties of weak solutions to the compressible isentropic Navier-Stokes equations in dimension 2 or 3. First, the restrictions on the pressure growth for the known global existence and stability results are illustrated through explicit radial solutions. On the other hand, an improved global time regularity is derived for the momentum.
Advances in Mathematics, 2021
In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the " quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.
Weak solutions to the equations of motion for compressible magnetic fluids
Journal de Mathématiques Pures et Appliquées, 2009
We study the differential system governing the flow of a compressible magnetic fluid under the action of a magnetic field. The system is a combination of the compressible Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. We prove global-in-time existence of weak solutions with finite energy to the system posed in a bounded domain of R 3 and equipped with initial and boundary conditions.