In this paper we are interested in the sufficient conditions which guarantee the regularity of solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval[O,T].Five sufficient couditions are gi...In this paper we are interested in the sufficient conditions which guarantee the regularity of solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval[O,T].Five sufficient couditions are given.Our results are motivated by two main ideas:one is to control the accumulation of vorticity alone;the other is to generalize the corresponding geometric conditions of 3-D Euler equations to 3-D ideal magnetohydrodynamic equations.展开更多
In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the cor...In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the corresponding Navier-Stokes equations, converge strongly in provided that they have strong convergence in the region away from the symmetry axis. This result has been proved by the authors for the approximate solutions generated by smoothing the initial data, with no restriction of the sign of the initial data. Then we discuss the decay rate for maximal vorticity function, which is established for both approximate solutions generated by smoothing the initial data and viscous approximations respectively. One sufficient condition to guarantee the strong convergence in the region away from the symmetry axis is given, and a decay rate for maximal vorticity function in the region away from the symmetry axis is obtained for non-negative initial vorticity.展开更多
Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = ux,x on [0,1], with the boundary condition u(0,t)=um,u(1t...Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = ux,x on [0,1], with the boundary condition u(0,t)=um,u(1t)=u+ and the initial data u(x,0)= u0(x, where um p u+ and f is a given function satisfying f'(u>0 for u under consideration. By means of energy estimates method and under some more regular conditions on the initial data, both the global existence and the asymptotic behavior are obtained. When um < u+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for um > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, which means that |um m u+| is small. Moreover, exponential decay rates are both given.展开更多
基金The first author is partially Supported by Natural sciences Foundation of china(No.10101014)Beijing Education Committee Foundation and the Key Project of NSFB-FBECThe second author is partially supported by Natural Sciences Foundation of China
文摘In this paper we are interested in the sufficient conditions which guarantee the regularity of solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval[O,T].Five sufficient couditions are given.Our results are motivated by two main ideas:one is to control the accumulation of vorticity alone;the other is to generalize the corresponding geometric conditions of 3-D Euler equations to 3-D ideal magnetohydrodynamic equations.
基金partially supported by National Natural Sciences Foundation of China (No.10101014)Beijing Natural Sciences Foundation+1 种基金the Key Project of NSFB-FBEC,by Grants from RGC of HKSAR CUHK4279/00P and CUHK4129/99Pthe generous hospitality and financial support of IMS of The Chinese University of Hong Kongpartially supported by Zheng Ge Ru Funds, Grants from RGC of HKSAR CUHK4279/00P and CUHK4129/99P
文摘In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the corresponding Navier-Stokes equations, converge strongly in provided that they have strong convergence in the region away from the symmetry axis. This result has been proved by the authors for the approximate solutions generated by smoothing the initial data, with no restriction of the sign of the initial data. Then we discuss the decay rate for maximal vorticity function, which is established for both approximate solutions generated by smoothing the initial data and viscous approximations respectively. One sufficient condition to guarantee the strong convergence in the region away from the symmetry axis is given, and a decay rate for maximal vorticity function in the region away from the symmetry axis is obtained for non-negative initial vorticity.
基金Partially supported by the National Natural Sciences Foundation of China (No. 10101014), the Key Project of Natural Sciences Foundation of Beijing and Beijing Education Committee Foundation.Supported by the National Natural Science Foundation of China
文摘Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = ux,x on [0,1], with the boundary condition u(0,t)=um,u(1t)=u+ and the initial data u(x,0)= u0(x, where um p u+ and f is a given function satisfying f'(u>0 for u under consideration. By means of energy estimates method and under some more regular conditions on the initial data, both the global existence and the asymptotic behavior are obtained. When um < u+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for um > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, which means that |um m u+| is small. Moreover, exponential decay rates are both given.
