论文标题
建造2D Euler方程的高规律性不变性措施，并就解决方案的增长进行评论
Construction of High Regularity Invariant Measures for the 2D Euler Equations and Remarks on the Growth of the Solutions
论文作者
论文摘要
我们考虑了二维圆环上的欧拉方程，并为这些方程的动力学构建不变的度量，集中在足够规则的Sobolev空间上，因此也已知存在强大的解决方案。证明遵循了库克辛的方法，我们特别获得这些措施没有原子，不包括琐碎的不变措施。然后，我们证明，几乎所有关于构造措施的初始数据都会引起全球解决方案，而Sobolev规范的增长最多是多项式的。为此，我们依靠一位波尔加恩的论点。 Kuksin's和Bourgain的论点的这种结合已经出现在SY的工作中。我们指出，符合作者的知识，这是Sobolev Norm to $ 2D $ EULER方程增长的唯一一般上限是双重指数。
We consider the Euler equations on the two-dimensional torus and construct invariant measures for the dynamics of these equations, concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist. The proof follows the method of Kuksin and we obtain in particular that these measures do not have atoms, excluding trivial invariant measures. Then we prove that almost every initial data with respect to the constructed measures give rise to global solutions for which the growth of the Sobolev norms are at most polynomial. To do this, we rely on an argument of Bourgain. Such a combination of Kuksin's and Bourgain's arguments already appears in the work of Sy. We point out that up to the knowledge of the author, the only general upper bound for the growth of the Sobolev norm to the $2d$ Euler equations is double exponential.
