稳定二维Boussinesq方程在$H^2$中不存在性：密度-涡量反馈机制

中文摘要 AI 辅助

我们建立了二维Boussinesq方程在线性稳定背景剖面$-x_2$附近的瞬时$H^2$范数爆炸。此前该系统的病态性结果依赖于$L^\infty$型空间，而本文提供了能量临界空间中的首个强病态性结果，其中稳定的背景分层抑制了形变增长。我们借鉴了先前关于IPM方程的工作\cite{BCMZ2025}中的思想，选取在原点附近消失的初始密度扰动以中和稳定分层。关键的是，与IPM情形不同，这里的演化由密度与涡量之间的非线性耦合驱动。由于初始涡量恒为零，瞬时正则性丧失由密度-涡量反馈驱动——这一机制与Bourgain和Li的$H^1$ Euler病态性以及IPM演化有根本区别。通过迭代方案生成一系列在原点具有强形变的近似解，我们构造了一个强解$(\rho(t), \omega(t))$，其初始数据具有任意小的$H^2 \times (H^1\cap \dot H^{-1})$范数，但立即对所有$t>0$表现出$H^2 \times H^1$范数爆炸，同时在能量类中保持唯一性。

英文摘要

We establish instantaneous $H^2$ norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile $-x_2$. While previous ill-posedness results for this system relied on $L^\infty$-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both $H^1$ Euler ill-posedness of Bourgain \& Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(ρ(t), ω(t))$ originating from data with arbitrarily small $H^2 \times (H^1\cap \dot H^{-1})$ norm, which immediately exhibits $H^2 \times H^1$ norm explosion for all $t > 0$ while maintaining uniqueness within the energy class.

发表机构

• Consiglio Nazionale Delle Ricerche（意大利国家研究委员会）
• University of Basel（巴塞尔大学）

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