Two-front solutions of the SQG equation and its generalizations
Two-front solutions of the SQG equation and its generalizations
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DOI：
10.4310/cms.2020.v18.n6.a8
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发表时间：
2019-04
期刊：
影响因子：
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通讯作者：
J. K. Hunter;Jingyang Shu;Qingtian Zhang
中科院分区：
文献类型：
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作者：
J. K. Hunter;Jingyang Shu;Qingtian Zhang
广义地表准地转 (GSQG) 方程是依赖于参数 $0<\alpha \le 2$ 的主动标量的输运方程。特殊情况是二维不可压缩欧拉方程 ($\alpha = 2$) 和表面准地转 (SQG) 方程 ($\alpha = 1$)。当前沿是图时，我们推导出 GSQG 方程的一类两前沿解的轮廓动力学方程。这些方程的标量简化包括描述存在刚性平坦边界的单个前沿的方程。我们使用轮廓动力学方程来确定对应于两个平坦锋面的 GSQG 剪切流的线性稳定性。我们还证明了参数范围 $1<\alpha\le 2$ 中两前沿方程的大的、光滑的解以及参数范围 $0<\alpha\le 1$ 中的小、光滑的解的局部时间存在性和唯一性。
The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<\alpha \le 2$. Special cases are the two-dimensional incompressible Euler equations ($\alpha = 2$) and the surface quasi-geostrophic (SQG) equations ($\alpha = 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<\alpha\le 2$, and small, smooth solutions in the parameter regime $0<\alpha\le 1$.