光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

Global Smooth Solutions With Exponential Growth to 2D Inviscid Boussinesq Equations Without Heat Conduction and 3D Axisymmetric Incompressible Euler Equations on Smooth Domains

研究二维无黏性无热传导 Boussinesq 方程组和三维轴对称不可压 Euler 方程组光滑解的增长情况,找各种区域使其上的方程组有快增长的解.对 Boussinesq 方程组,通过选取初始温度和速度的一个分量,可以把方程去耦为两部分.从关于涡量的部分求出涡量、速度场和使结论成立的区域,从关于温度的部分可见,温度的高阶导的增长仅依赖于速度场的一个分量.通过适当选取该分量,得到温度高阶导有指数增长的全局光滑解.对轴对称 Euler 方程组做类似的处理,适当选取速度场的径向分量,可把方程组去耦,最终得到一类光滑区域,在其上方程组有指数增长全局光滑解.该研究把 Chae、Constantin、Wu 对一个二维锥形区域上无黏性无热传导 Boussinesq 方程的结果,推广到一类光滑区域上,并把他们的方法应用到三维轴对称不可压 Euler 方程组,得到了类似的结果.

The growth of smooth solutions to 2D inviscid Boussinesq equations without heat conduction and the 3D axisymmetric Euler equations was investigated,to find regions where these systems have fast growing solutions. Through appropriately choosing the initial temperature and the velocity component,the Boussinesq system was decoupled into 2 parts. From the part involving only the vorticity,the vorticity and velocity can be solved and the smooth regions determined. From the part involving the temperature,one can see that the growth of temperature derivatives depends only on the velocity component. Through choosing that component appropriately,solutions with temperature derivatives of exponential growth were constructed on certain unbound smooth regions. The same method was applied to the axisymmetric Euler equations. Through choosing the radial velocity component appropriately,the system can be decoupled and one can ultimately find a class of smooth domains,and on them smooth global solutions of exponential growth. This investigation extends the results of Chae,Constantin and Wu on the inviscid Boussinesq system without heat conduction on a 2D cone to a class of smooth domains. Their method was also applied to the 3D axisymmetric Euler equations to obtain a similar result.