This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtained for the quasi-geostrophic equation.

中文翻译：

---

具有临界耗散的Burgers方程和拟地转方程的解析性和大时间行为。

本文涉及具有临界耗散的Burgers方程的柯西问题。基于频率分解方法，研究了空间和时间变量的适定性和解析性。对于任何较大的初始数据，都会显示较大的时间行为。结果表明，只要时间为正，任何光滑且可积分的解决方案都可以在空间和时间上进行分析，并且随着时间趋于无穷大，其行为类似于泊松核。对于准地转方程也获得了相应的结果。