354 research outputs found
Singularity formation to the Cauchy problem of the two-dimensional non-baratropic magnetohydrodynamic equations without heat conductivity
We study the singularity formation of strong solutions to the two-dimensional
(2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic
equations without heat conductivity. It is proved that the strong solution
exists globally if the density and the pressure are bounded from above. In
particular, the criterion is independent of the magnetic field and is just the
same as that of the compressible Navier-Stokes equations. Our method relies on
weighted energy estimates and a Hardy-type inequality.Comment: 19 pages. arXiv admin note: substantial text overlap with
arXiv:1705.05161, arXiv:1705.06606; text overlap with arXiv:1801.0758
Blow-up criterion for the D non-resistive compressible Magnetohydrodynamic equations
In this paper, we prove a blow-up criterion in terms of the magnetic field
and the mass density for the strong solutions to the D
compressible isentropic MHD equations with zero magnetic diffusion and initial
vacuum. More precisely, we show that the norms of control
the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which
means that if a solution of the compressible isentropic non-resistive MHD
equations is initially smooth and loses its regularity at some later time, then
the formation of singularity must be caused by losing the bound of the
norm of or as the critical time approaches.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1401.270
A blow-up criterion of strong solutions to the 2D compressible magnetohydrodynamic equations
This paper establishes a blow-up criterion of strong solutions to the
two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion
depends on the density, but is independent of the velocity and the magnetic
field. More precisely, once the strong solutions blow up, the -norm
for the density tends to infinity. In particular, the vacuum in the solutions
is allowed.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1402.4851,
arXiv:1310.1673, arXiv:1210.5930 by other author
A blow-up criterion for strong solutions to three-dimensional compressible magnetohydrodynamic equations
We are concerned with an initial boundary value problem for the compressible
magnetohydrodynamic equations with viscosity depending on the density. It is
show that for the initial density away from vacuum, the strong solution to the
problem exists globally if the gradient of velocity satisfies
. Our method relies upon the
delicate energy estimates and elliptic estimates.Comment: 17 page
On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction
We are concerned with the formation of singularity and breakdown of strong
solutions to the Cauchy problem of the three-dimensional full compressible
magnetohydrodynamic equations with zero heat conduction. It is proved that for
the initial density allowing vacuum, the strong solution exists globally if the
deformation tensor and the pressure satisfy
.
In particular, the criterion is independent of the magnetic field. The
logarithm-type estimate for the Lam{\'e} system and some delicate energy
estimates play a crucial role in the proof.Comment: to appear in Indiana University Mathematics Journal. arXiv admin
note: text overlap with arXiv:1705.0516
Singularity formation to the 2D Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction
The formation of singularity and breakdown of strong solutions to the
two-dimensional (2D) Cauchy problem of the full compressible Navier-Stokes
equations with zero heat conduction are considered. It is shown that for the
initial density allowing vacuum, the strong solution exists globally if the
density and the pressure satisfy
.
In addition, the initial density can even have compact support. The
logarithm-type estimate for the Lam{\'e} system and some weighted estimates
play a crucial role in the proof.Comment: 15 page
Global classical solution to 3D compressible magnetohydrodynamic equations with large initial data and vacuum
In this paper, we study the Cauchy problem of the isentropic compressible
magnetohydrodynamic equations in . When
, together with the
, is suitably small, a result on the existence of global
classical solutions is obtained. It should be pointed out that the initial
energy except the - norm of can be large as
goes to 1, and that throughout the proof of the theorem in the present paper,
we make no restriction upon the initial data . Our result
improves the one established by Li-Xu-Zhang in \cite{H.L. L}, where, with small
initial engergy, the existence of classical solution was proved.Comment: 36 page
Global well-posedness and large time asymptotic behavior of strong solutions to the 2-D compressible magnetohydrodynamic equations with vacuum
The authors study the Cauchy problem of the magnetohydrodynamic equations for
viscous compressible barotropic flows in two or three spatial dimensions with
vacuum as far field density. For two spatial dimensions, we establish the
global existence and uniqueness of strong solutions (which may be of possibly
large oscillations) provided the smooth initial data are of small total energy,
and obtain some a priori decay with rates (in large time) for the pressure, the
spatial gradient of both the velocity field and the magnetic field. Moreover,
for three spatial dimensions case, some similar decay rates are also obtained.Comment: arXiv admin note: substantial text overlap with arXiv:1310.1673,
arXiv:1004.4749, arXiv:1207.3746, arXiv:1107.4655 by other author
On classical solutions to the Cauchy problem of the 2D compressible non-resistive MHD equations with vacuum
In this paper, we investigate the Cauchy problem of the compressible
non-resistive MHD on with vacuum as far field density. We prove
that the 2D Cauchy problem has a unique local strong solution provided the
initial density and magnetic field decay not too slow at infinity. Furthermore,
if the initial data satisfies some additional regularity and compatibility
conditions, the strong solution becomes a classical one. Additionally, we
establish a blowup criterion for the 2D compressible non-resistive MHD
depending solely on the density and magnetic fields.Comment: To appear in Nonlinearity. arXiv admin note: text overlap with
arXiv:1707.05279; and text overlap with arXiv:1306.4752, arXiv:1506.02156 by
other author
Exponential Decay for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations in 3D Bounded Domains
For barotropic compressible Navier-Stokes equations in three-dimensional (3D)
bounded domains, we prove that any finite energy weak solution obtained by
Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible
models(1998)] and Feireisl-Novotn\'{y}-Petzeltov\'{a} [J. Math. Fluid Mech.
3(2001), 358-392] decays exponentially to the equilibrium state. This result is
established by both using the extra integrability of the density due to Lions
and constructing a suitable Lyapunov functional just under the framework of
Lions-Feireisl's weak solutions.Comment: 16 page
- β¦