We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.
This paper is concerned with a blow-up criterion of strong solutions for three-dimensional compressible isentropic magnetohydrodynamic equations with vacuum. It is shown that if the density and velocity satisfy [Formula: see text], where [Formula: see text], 3 < r ≤ ∞ and [Formula: see text] denotes the weak Lr-space, then the strong solutions to the Cauchy problem of the compressible magnetohydrodynamic equations can exist globally over [0, T].
In this paper we investigate the Cauchy problem for the three-dimensional incompressible magnetohydrodynamic equations. A logarithmal improved blow-up criterion of smooth solutions is obtained.
On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in $${\mathbb{R}^{3}}$$ R 3
