论文标题
尺寸不可压缩的Euler方程的准周期溶液两个及更高
Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher
论文作者
论文摘要
在Crouseilles和Faou的工作中，我们在2D案例上的工作，为不可压缩的Euler方程构建了$ C^\ Infty $ Quasi-periodic解决方案，并在尺寸3和任何均匀的尺寸中定期边界条件。这些解决方案是真正的高维度，这特别有趣，因为很少有高维初始数据的例子已知全球解决方案存在。这些准周期的溶液可以设计，以便它们在嵌入螺线管矢量场空间中的任意维尺寸的摩tori上密集。此外，在二维情况下，我们表明准周期溶液在Euler方程的相空间中密集。更准确地说，对于任何整数$ n \ geq 1 $，我们证明，任何$ l^q $初始流函数都可以在$ l^q $中近似（强烈的时候，当$ 1 \ leq q <\ infty $和nef-iffty $和nef-*当$ q = \ infty $）通过平滑的初始数据在$ n $ n $ dimemensional上加密的求解。
Building on the work of Crouseilles and Faou on the 2D case, we construct $C^\infty$ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer $N\geq 1$ we prove that any $L^q$ initial stream function can be approximated in $L^q$ (strongly when $1\leq q< \infty$ and weak-* when $q=\infty$) by smooth initial data whose solutions are dense on $N$-dimensional tori.
