论文标题
解决方案的解决方案的存在和独特性在Sobolev空间中
Existence and Uniqueness of the Solution to the Anisotropic Quasi-Geostrophic Equations in the Sobolev Space
论文作者
论文摘要
在本文中，我们专注于具有分数水平耗散和垂直热扩散的二维表面准地藻方程，该方程代表了经典表面准地缘方程的一般情况。一方面，我们将在sobolev空间中展示解决方案的局部存在和独特性，$ h^{2-2α}（\ mathbb {r}^2）\ cap h^{2-2β}（\ mathbb {r {r}^2）$，这在经典案例中是至关重要的空间。此外，我们将证明即使初始数据很小，该解决方案也是全局的。最后，我们将研究我们无限全球解决方案的渐近表示。
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of our global solution in infinity.
