受控调和映射热流及从圆盘到球面的爆破

中文摘要 AI 辅助

我们研究从圆盘到球面的单余旋调和映射热流，并带有Dirichlet边界控制。我们引入一种适应控制的粘合策略来阻止爆破。更确切地说，我们构造了显式的上界轮廓和良好预备的边界迹，使得任何非负数据若低于这些轮廓之一且边界值高于$\pi$，则在相应的常数迹下会爆破，而对应的解在低于相关良好预备边界迹的每个控制下是全局的。具有一致有界时间导数的控制也能阻止无限时间浓度。我们还表明，边界控制可以在任何预定时间之前驱动任何非负数据发生爆破，而有界控制无法阻止适当快速的浓度。

英文摘要

We study the one-corotational harmonic map heat flow from the disk to the sphere with a Dirichlet boundary control. We introduce a control-adapted gluing strategy to prevent blow-up. More precisely, we construct explicit upper profiles and well-prepared boundary traces such that every nonnegative datum below one of these profiles and having boundary value above $π$ blows up under the corresponding constant trace, whereas the corresponding solution is global under every control below the associated well-prepared boundary trace. Controls with uniformly bounded time derivative also prevent infinite-time concentration. We also show that boundary control can drive any nonnegative datum to blow-up before any prescribed time, whereas bounded controls cannot prevent suitably fast concentration.

发表机构

• School of Mathematical Sciences, Peking University（北京大学数学科学学院）

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