Narrow Results

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.

In this short note, we consider smooth solutions to certain hyperbolic systems of equations. We present a condition which will ensure that no shocks develop and that solutions decay in L². The condition is restrictive in general; however, when applied to the system of one-dimensional gas dynamics it is shown that if the condition is satisfied initially then it will be satisfied for all time and therefore one obtains smooth solutions which decay.

The Einstein equations with a positive cosmological constant are coupled to the pressureless perfect fluid matter in plane symmetry. Under suitable restrictions on the initial data, the resulting Einstein-dust system is proved to have a global classical solution in the future time direction. Some late time asymptotic properties are obtained as well.

In this paper, we investigate the well-posedness for the Euler–Boltzmann equations of radiation hydrodynamics in one spatial variable. We obtain the local existence and uniqueness of smooth solution to the initial-boundary value problem. Then, we show that a smooth solution will blow up in finite time regardless of the size of the initial disturbance.

In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line $[0,+\infty )$ for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line $[0,+\infty )$ by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval $[0,1]$ by introducing some new average quantities.