周期Benjamin-Bona-Mahony方程的唯一延拓
Unique continuation for the periodic Benjamin-Bona-Mahony equation
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中文总结 AI 辅助
本文证明了能量空间中周期Benjamin-Bona-Mahony方程的唯一延拓，无需小性或均值假设，并推广至任意实多项式通量，从而无条件确立Rosier的镇定结果，同时指出定理在低正则性空间中的失效。
中文摘要 AI 辅助
我们证明了能量空间中实周期Benjamin-Bona-Mahony方程的唯一延拓：若一个解在非空时空条带上等于常数，则该解处处等于该常数。特别地，Rosier的唯一延拓猜想[23]在没有任何小性条件或均值假设的情况下成立。甚至只需在一点$x_0$处且$t$属于某个区间时，观测解及其通量的势$\u03c8=(1-\u2202_x^2)^{-1}(u+u^2/2)$：$u(x_0,t)=C$且$\u03c8(x_0,t)=C+C^2/2$即可推出$u\equiv C$，且当$C=-1$时第二个条件不可省略。证明使用了具有不定号的指数加权Hamiltonian以及耗散恒等式，该方法适用于任意非常数实多项式通量。因此，Rosier关于两个局部反馈律和边界反馈的镇定结果在任意数据下无条件成立。稳态阶梯剖面表明该定理在$0\leq s<1/2$的$H^s$空间中不成立。我们还给出了线性化方程单点唯一延拓的生成级数证明，并讨论了相关的控制结果。
英文摘要
We prove unique continuation for the real periodic Benjamin--Bona--Mahony equation in the energy space: a solution that equals a constant on a nonempty space--time strip is that constant everywhere. In particular, Rosier's unique continuation conjecture [23] holds without any smallness or mean assumption. It even suffices to observe, at one point $x_0$ and for $t$ in an interval, the solution and the potential $ψ=(1-\partial_x^2)^{-1}(u+u^2/2)$ of its flux: $u(x_0,t)=C$ and $ψ(x_0,t)=C+C^2/2$ imply $u\equiv C$, and the second condition cannot be dropped when $C=-1$. The proof uses an exponentially weighted Hamiltonian of indefinite sign with a coercive dissipation identity, and it applies to every nonconstant real polynomial flux. Consequently, Rosier's stabilization results for two localized feedback laws and for a boundary feedback hold unconditionally, for arbitrary data. Stationary step profiles show that the theorem fails in $H^s$ for $0\leq s<1/2$. We also give a generating-series proof of single-point unique continuation for the linearized equation and discuss related control results.
发表机构
- Universidade Federal de Pernambuco (UFPE)（伯南布哥联邦大学）
- Université Sorbonne Paris-Nord（索邦巴黎北大学）
- Universidade Federal do Piauí (UFPI)（皮奥伊联邦大学）
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