论文标题
$ l^\ infty $ in Riesz Transform问题的强大不良
Strong Ill-Posedness in $L^\infty$ for the Riesz Transform Problem
论文作者
论文摘要
对于形式的2D Euler方程的线性扰动，我们证明了$ l^{\ infty} $中的强大不良性：\ [\ [\partial_tΩ+ cdot \ cdot \ cdot \nablaΩ= r（ω），\]，其中$ r $是任何非案件的第二阶ries ries ries ries ries riesz transils $ r $。也就是说，我们证明存在$ l^{\ infty} $中最初很小的平滑解决方案，但在短时间内任意变大。以前朝这个方向的工作依赖于线性问题的强大不良性，从而扰动地观察了运输术语，这仅导致了轻微的生长。在这项工作中，我们得出了一个非线性模型，考虑了所有领先的顺序效应，以在短时间内确定解决方案的精确增长。有趣的是，Euler运输项确实抵消了线性生长，因此整个非线性方程的生长比线性的数量级要小。特别是，我们确定的（尖锐的）增长率与平滑解决方案的全球规律性一致。
We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t ω+ u\cdot\nablaω= R(ω),\] where $R$ is any non-trivial second order Riesz transform. Namely, we prove that there exist smooth solutions that are initially small in $L^{\infty}$ but become arbitrarily large in short time. Previous works in this direction relied on the strong ill-posedness of the linear problem, viewing the transport term perturbatively, which only led to mild growth. In this work we derive a nonlinear model taking all of the leading order effects into account to determine the precise pointwise growth of solutions for short time. Interestingly, the Euler transport term does counteract the linear growth so that the full nonlinear equation grows an order of magnitude less than the linear one. In particular, the (sharp) growth rate we establish is consistent with the global regularity of smooth solutions.
