边界处的测度与连续向量场 II：测地线与支集传播
Measure and continuous vector field at a boundary II: geodesics and support propagation
AI总结：
本文研究带边界区域上连续向量场输运方程的非负测度，通过定义遵循几何光学的广义双特征线，证明其支集包含于极大广义双特征线并集，并用于完成低正则性波动方程能观性证明。
摘要：
作为具有连续系数的输运方程解的非负测度已得到广泛研究。由于相关向量场的正则性较低，不存在自然流，因为积分曲线的非唯一性是普遍规律。自 L. Ambrosio [2] 以及 L. Ambrosio 和 G. Crippa [3, 4] 的工作以来，人们已经知道此类测度可以描述为支撑在积分曲线上的 $δ$-测度的叠加。在本文中，受波动方程的一些能观性问题启发，我们关注带边界区域情形下的此类输运方程。与具有 $C^1$ 系数的波动方程相关联的是双特征线，它们是连续 Hamilton 向量场的积分曲线。我们首先详细研究它们在边界存在时的行为，并定义遵循几何光学定律的自然推广。随后，我们引入一类自然的、在边界上带有源项的输运方程，并证明任何满足此类方程的非负测度都以极大广义双特征线的并集为支集。该结果是边界存在时叠加原理的一种弱形式。结合其姊妹篇 [7]，本研究完成了波动方程能观性的证明，在波动方程（及相关度量）系数为 $C^1$、边界和流形为 $C^2$ 的低正则性框架中，推广了 Bardos、Lebeau 和 Rauch [5] 的著名结果。
英文摘要：
Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral curves is the general rule. It has been known since the works by L. Ambrosio [2] and L. Ambrosio and G. Crippa [3, 4] that such measures can be described as a superposition of $δ$-measures supported on integral curves. In this article, motivated by some observability questions for the wave equation, we are interested in such transport equations in the case of domains with boundary. Associated with a wave equation with C^1-coefficients are bicharacteristics that are integral curves of a continuous Hamiltonian vector field. We first study in details their behaviour in the presence of a boundary and define their natural generalisation that follows the laws of geometric optics. Then, we introduce a natural class of transport equations with a source term on the boundary, and we prove that any nonnegative measure satisfying such an equation has a union of maximal generalized bicharacteristics for support. This result is a weak form of the superposition principle in the presence of a boundary. With its companion article [7], this study completes the proof of wave observability generalizing the celebrated result of Bardos, Lebeau, and Rauch [5] in a low regularity framework where coefficients of the wave equation (and associated metric) are C^1 and the boundary and the manifold are C^2.
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