AI 中文总结
研究二维Boussinesq系统在超临界Besov空间的范数膨胀，通过分析无粘和全耗散系统在特定条件下的情况，得出范数膨胀发生在密度分量\(\rho\)且速度分量\(u\)有界等结论，覆盖多数满足条件的超临界空间。
AI 中文摘要
我们证明了二维Boussinesq系统在超临界Besov空间中存在强不适定性意义下的范数膨胀。对于无粘系统，在\(\dot B^\beta_{p,q}(\mathbb R^2)\times \dot B^\beta_{p,r}(\mathbb R^2)\)中当\(\beta\neq0\)，\(1<p\leq\infty\)，\(1\leq q,r\leq\infty\)且\(-2<\beta-\frac{2}{p}<1\)时范数膨胀成立；对于全耗散系统，在\(-2<\beta-\frac{2}{p}<-1\)范围内同样结论成立。两种情况结果覆盖几乎所有满足局部可积条件的超临界Besov空间。范数膨胀发生在密度分量\(\rho\)，速度分量\(u\)保持有界。在全耗散情形，膨胀空间对\(u\)是超临界的，对\(\rho\)是亚临界的。
英文摘要
We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in \(\dot B^β_{p,q}(\mathbb R^2)\times \dot B^β_{p,r}(\mathbb R^2)\) for \(β\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), and \(-2<β-\frac{2}{p}<1\). For the fully dissipative system, the same conclusion holds in the range \(-2<β-\frac{2}{p}<-1\). In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component \(ρ\), while the velocity component \(u\) remains bounded. In the fully dissipative case, the inflation space is supercritical for \(u\), but subcritical for \(ρ\) with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in \(ρ\).
Comments35 pages
