In this paper, we prove local existence and uniqueness of analytic sharp-front solutions to a generalised SQG equation by the use of an abstract Cauchy–Kovalevskaya theorem. Here, the velocity is determined by $u={|\nabla |}^{-2\beta }{\nabla }^{\perp }\theta$ which (for $1<\beta \le 2$) is more singular than in SQG. This is achieved despite the appearance of pseudodifferential operators of order higher than one in our equation, by recasting our equation in a suitable integral form. We also provide a full proof of the abstract version of the Cauchy–Kovalevskaya theorem we use.

中文翻译：

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奇异SQG解析锋线的局部存在

在本文中，我们通过使用抽象的柯西-科瓦列夫斯卡娅定理证明了广义SQG方程的解析尖锐前沿解的局部存在性和唯一性。在这里，速度由$ü={|\nabla |}^{-2\beta }{\nabla }^{\perp }\theta$ 哪个（对于 $\mathrm{1个}<\beta \le 2$）比SQG中的单数更奇。尽管出现了伪微分算子的阶数高于我们的等式，但仍可以通过以合适的积分形式重铸我们的等式来实现。我们还提供了所用柯西–科瓦列夫斯卡夫定理的抽象版本的完整证明。