Cusp Formation for a Nonlocal Evolution Equation
Cusp Formation for a Nonlocal Evolution Equation
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非局部演化方程的尖点形成
DOI：
10.1007/s00205-017-1094-3
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发表时间：
2016
影响因子：
2.5
通讯作者：
M. Radosz
中科院分区：
文献类型：
--
作者：
V. Hoang;M. Radosz
科尔多瓦等人(Ann Math 162（3）：1377-1389，2005）引入了一个非局部活动标量方程作为表面准地转方程的一维模拟。它已被证明，基于数值证据，该解决方案在有限时间内形成一个尖状奇点。到目前为止，还没有一个具有非局部通量的活动标量被严格地证明了尖点的形成。在本文中，我们引入并研究了一个非局部的活动标量，启发Córdoba-Córdoba-Fontelos方程，并证明了尖状或针状奇点的形式在有限时间。
Córdoba et al. (Ann Math 162(3):1377–1389, 2005) introduced a nonlocal active scalar equation as a one-dimensional analogue of the surface-quasigeostrophic equation. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for which cusp formation has been rigorously shown. In this paper, we introduce and study a nonlocal active scalar, inspired by the Córdoba–Córdoba–Fontelos equation, and prove that either a cusp- or needle-like singularity forms in finite time.