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.

中文翻译：

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修正的 Michelson-Sivashinsky 方程的强解

我们证明了在任何空间维度和没有物理边界的情况下对稍微修改的 Michelson-Sivashinsky 方程的强解的全局适定性和规律性结果。空间 $W^{1,\infty} (\mathbb{R}^d)$ 中的局部时间适定性（和规律性）被建立并且如果另外初始数据是周期性或无穷远处消失。后一个结果的证明利用了先前由 Kiselev、Nazarov、Volberg 和 Shterenberg 引入的思想来处理临界耗散表面准地转方程和临界耗散分数 Burgers 方程。即，全局规律性结果是通过构造一个时间相关的连续性模数来实现的，该连续性模数必须始终被初值问题的解所遵守，防止溶液梯度膨胀。这项工作提供了一个例子，即使在先验界限不可用。