圆盘上有外力不可压Euler方程组光滑解的增长

Growth of smooth solutions of the incompressible Euler equations with external force on the disk

证明圆盘上面有外力不可压Euler方程组的光滑解的涡量梯度对时间可以达到二阶幂指数增长.对无外力情况已经得到同样的结果.在有外力的情况下,要更小心地对速度场作估计才能得到结论.有外力不可压Euler方程组跟无粘性无热传导Boussinesq方程组有相似之处,其中的涡量方程都有外力项,希望通过研究前者得到研究后者的方法启示.

We proved that for the incompressible Euler equations with external force on the disk, there was a smooth solution with vorticity gradient growing double exponentially. The result had been proved for the case without external force. We need to estimate the velocity field more carefully to get the same result when external force is present. The incompressible Euler equations with external force is similar to the inviscid Boussinesq system without heat conduction in both systems,the vorticity equation has a force term. Investigating the former may throw light on how to investigate the latter.