不可压缩多孔介质方程在Lei-Lin空间中解的全局存在性和解析性

Global Existence and Analyticity of Solution of Incompressible Porous Media Equation in Lei-Lin Spaces

摘要: 本文考虑二维不可压缩多孔介质方程的Cauchy问题。首先通过Fourier变换将原方程转化为等价积分方程，随后在Lei-Lin型函数空间框架下，系统分析了线性项与非线性项的估计。基于Banach动点定理，我们证明${\Vert {\theta }_0\Vert }_{{\mathcal{X}}^{1-2s}}$ 充分小时，方程存在唯一的全局解。此外，进一步给出了方程解的解析性。

Abstract: This paper investigates the Cauchy problem for the two-dimensional incompressible porous medium equation. First, we transform the original equation into an equivalent integral equation via the Fourier transform. Subsequently, within the framework of Lei-Lin-type function spaces, we systematically analyze the estimates of both the linear and nonlinear terms. Based on the Banach fixed-point theorem, we prove that when the initial data ${\Vert {\theta }_0\Vert }_{{\mathcal{X}}^{1-2s}}$ is sufficiently small, there exists a unique global solution to the equation. Furthermore, we establish the analyticity of the solution.