端点Sobolev空间中不可压缩欧拉方程的整体适定性
Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space
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中文总结 AI 辅助
研究端点临界Sobolev空间中涡度方程初值问题，在二维证明涡度\(W^{2,1}\)正则性整体传播，三维中对无旋轴对称流证明涡度\(W^{3,1}\)正则性全时传播，与现有临界Sobolev空间强不适定性结果不同。
中文摘要 AI 辅助
我们考虑在端点临界Sobolev空间\(W^{d,1}(\mathbb{R}^{d})\)（\(d = 2, 3\)）中涡度方程的初值问题。在二维中，我们证明了涡度\(W^{2,1}(\mathbb{R}^{2})\)正则性的整体传播。在三维中，对于无旋轴对称流，我们证明了涡度\(W^{3,1}(\mathbb{R}^{3})\)正则性在所有时间的传播。这与\(1 < p < \infty\)时\(W^{d/p,p}(\mathbb{R}^{d})\)临界Sobolev空间中现有的强不适定性结果形成鲜明对比。
英文摘要
We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space $W^{d,1}(\mathbb{R}^{d})$ for $d = 2, 3$. In two dimensions, we prove global propagation of the $W^{2,1}(\mathbb{R}^{2})$ regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate $W^{3,1}(\mathbb{R}^{3})$ regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces $W^{d/p,p}(\mathbb{R}^{d})$ for all $1 < p < \infty$, which were based on axisymmetric flows without swirl when $d = 3$.
