发表机构
Università degli Studi Roma Tre; International School for Advanced Studied (SISSA)（罗马第三大学; 国际高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究重力与表面张力作用下二维水波的可控性，针对一般底部地形（开稠密集合）的无旋波和平底情形的恒定涡量波，分别证明了可控性结果，并克服了特征值非显式带来的技术困难。
AI 中文摘要
本文研究在重力和表面张力作用下的二维水波。我们证明了在有限深度且具有一般底部地形的流体域中，无旋波的可控性结果。该结果适用于$H^{s+1/2}(\mathbb{T})$（其中$s$足够大）中一个开且稠密的底部地形集合，这些地形不接触自由表面。我们指出，我们所允许的底部地形不一定是平底情形的小扰动：这导致了许多技术困难，因为对于一般底部地形，静止自由表面上的Dirichlet-Neumann算子的特征值不是显式的，也不一定（对于低频）接近平底情形下相应算子的特征值。反过来，这导致需要更复杂的论证来证明Ingham型估计，这些估计是证明可观测性所必需的，并促使我们限制上述可允许的底部地形。我们还证明了在具有平底地形的流体域中，具有恒定涡量的波的可控性结果。
英文摘要
In this paper we consider two-dimensional water waves, under the action of gravity and surface tension. We prove a controllability result for irrotational waves in a fluid domain with finite depth and general bottom topography. The result holds for an open and dense set of bottom topographies in $H^{s+1/2}(\mathbb{T})$ (where $s$ is sufficiently large) that do not touch the free surface. We point out that the bottom topographies that we allow for are not necessarily small perturbations of the flat bottom case: this leads to many technical difficulties, since the eigenvalues of the Dirichlet-Neumann operator at a still free surface with general bottom topography are not explicit, nor are they necessarily close (for low frequencies) to the eigenvalues of the corresponding operator for the flat bottom case. In turn, this leads to a more involved argument to prove Ingham-type estimates, which are needed to prove observability, and motivates the restriction mentioned above on the admissible bottom topographies. We also prove a controllability result for waves with constant vorticity in a fluid domain with flat bottom topography.
