论文标题
改进的米歇尔森 - 摩西山等方程的强大解决方案
Strong solutions to a modified Michelson-Sivashinsky equation
论文作者
论文摘要
在任何空间维度和没有物理边界的情况下，我们证明了全球良好的和规律性的结果。在空间中的本地时间良好（和规律性）$ w^{1，\ infty}（\ mathbb {r}^d）$被建立，如果此外，如果初始数据在Infinity处于周期性或消失，则显示为全局。后一个结果的证明利用了Kiselev，Nazarov，Volberg和Shterenberg先前提出的想法来处理急剧耗散的表面准氏藻方程以及急剧耗散的分数汉堡方程。也就是说，通过构建时间依赖性的连续性模量，可以通过有史以来的初始值问题解决方案来遵守，从而实现了整体规则性结果，从而阻止了溶液梯度的爆炸。这项工作提供了一个示例，即使没有A-Priori界限，也证明了规律性持续存在。
We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty}(\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative surface quasi-geostrophic equation and the critically dissipative fractional Burgers equation. Namely, the global regularity result is achieved by constructing a time-dependent modulus of continuity that must be obeyed by the solution of the initial-value problem for all time, preventing blowup of the gradient of the solution. This work provides an example where regularity is shown to persist even when a-priori bounds are not available.
