Search

We are concerned with the uniform regularity estimates and vanishing viscosity limit of solution to two dimensional viscous compressible magnetohydrodynamics (MHD) equations with transverse background magnetic field. When the magnetic field is assumed to be transverse to the boundary and the tangential component of magnetic field satisfies zero Neumann boundary condition, even though the velocity is imposed the no-slip boundary condition, the uniform regularity estimates of solution and its derivatives still can be achieved in suitable conormal Sobolev spaces in the half plane $\mathbb{R}^2_+$, and then the vanishing viscosity limit is justified in $L^\infty$ sense based on these uniform regularity estimates and some compactness arguments. At the same time, together with \cite{CLX21}, our results show that the transverse background magnetic field can prevent the strong boundary layer from occurring for compressible magnetohydrodynamics whether there is magnetic diffusion or not., Comment: 33 pages. arXiv admin note: text overlap with arXiv:2108.12969

In this paper, we establish $\varepsilon$-regularity criteria at one scale for suitable weak solutions to the five dimensional stationary incompressible Navier-Stokes equations in both the unit ball $B_1$ and the unit half ball $B_1^+$, respectively, which improve previous results in [J. Math. Fluid Mech., 6(2004), pp 78--101] and [Comm. Pure Appl. Math., 41(1988), pp 437--458]. The proofs of the main results are based on an iteration argument, Campanato's method, and interpolation techniques.

We are concerned with the uniform regularity estimates of solutions to the two dimensional compressible non-resistive magnetohydrodynamics (MHD) equations with the no-slip boundary condition on velocity in the half plane. Under the assumption that the initial magnetic field is transverse to the boundary, the uniform conormal energy estimates are established for the solutions to compressible MHD equations with respect to small viscosity coefficients. As a direct consequence, we proved the inviscid limit of solutions from viscous MHD systems to the ideal MHD systems in $L^\infty$ sense. It shows that the transverse magnetic field can prevent the boundary layers from occurring in some physical regime., Comment: 57 pages

In this paper, we consider the nonlinear inhomogeneous compressible elastic waves in three spatial dimensions when the density is a small disturbance around a constant state. In homogeneous case, the almost global existence was established by Klainerman-Sideris [1996_CPAM], and global existence was built by Agemi [2000_Invent. Math.] and Sideris [1996_Invent. Math., 2000_Ann. Math.] independently. Here we establish the corresponding almost global and global existence theory in the inhomogeneous case.