论文标题
分层溶液对IPM方程的松弛近似和渐近稳定性
Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation
论文作者
论文摘要
我们证明了稳定分层的解决方案对不可压缩的多孔介质方程（IPM）的非线性渐近稳定性，用于在$ \ dot H^{1- {1-τ}（\ Mathbb {r}^2）中的扰动（\ cap \ cap \ cap \ dot h^s（\ mathbb {r}^2）（\ cap \ dot h^s）这样的结果改善了现有文献，其中证明了至少属于$ h^{20}（\ Mathbb {r}^2）$的初始扰动证明了渐近稳定性。更确切地说，本文的目的是三倍。首先，我们提供了简化和改进的BousSinesQ方程式全球及时及时的证明，并在$ H^{1-τ}（\ Mathbb {r}^2）\ cap \ dot h^s（\ mathbb {\ mathb {r}^2）$中，$ h^{1-τ}（\ Mathbb {r}^2）$ s> 3 $ 3 $ 0 <1 $ 0 $ 0 $ 0.接下来，我们证明了在合适的缩放下，BousSinesQ系统具有阻尼涡度的强烈收敛。最后，分层溶液对（IPM）的渐近稳定性是副产品。通过各向异性Littlewood-paley分解的近似系统的对称性和对方程的各向异性特性的仔细研究起着关键作用，以获得均匀的能量估计。最后，主要的新事物和关键点之一是垂直速度$ \ | | U_2（t）\ | _ {l^\ infty（\ Mathbb {r}^2）} $的初始数据仅在$ \ dot H^{1- {1- {1- \ Mathbb {\ Mathbb {R}^2）中， h^s（\ mathbb {r}^2）$，$ s> 3 $。
We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot H^{1-τ}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and for any $0 < τ<1$. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to $H^{20}(\mathbb{R}^2)$. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in $H^{1-τ}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and $0 < τ<1$. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity $\|u_2(t)\|_{L^\infty (\mathbb{R}^2)}$ for initial data only in $\dot H^{1-τ}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s >3$.
