Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection
Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection
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DOI：
10.1080/03605302.2021.1975131
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发表时间：
2020-09
影响因子：
1.9
通讯作者：
Yuanyuan Feng-;A. Mazzucato
中科院分区：
文献类型：
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作者：
Yuanyuan Feng-;A. Mazzucato
摘要：我们通过不可压缩矢量场研究了有或没有平流的二维圆环上标量形式的 Kuramoto-Sivashinsky 方程 (KSE)。我们证明了 L 2 中任意数据的温和解的局部存在性。然后我们研究了全局存在性问题。我们证明了在任意数据存在平流的情况下 KSE 的全局存在性，前提是平流速度场 v 满足某些条件，确保相关超扩散-平流方程的耗散时间足够小。在没有平流的情况下，只有当线性算子不允许任何增长模式并且初始数据足够小时，才能显示全局存在。
Abstract We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on the two-dimensional torus with and without advection by an incompressible vector field. We prove local existence of mild solutions for arbitrary data in L 2. We then study the issue of global existence. We prove global existence for the KSE in the presence of advection for arbitrary data, provided the advecting velocity field v satisfies certain conditions that ensure the dissipation time of the associated hyperdiffusion-advection equation is sufficiently small. In the absence of advection, global existence can be shown only if the linearized operator does not admit any growing mode and for sufficiently small initial data.