从部分数据中恢复对流扩散方程的系数
Recovery of coefficients for a convection-diffusion equation from partial data
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中文总结 AI 辅助
研究从部分边界测量中恢复抛物型方程中依赖时空变量的零阶和一阶系数的反问题，通过开发特殊解、引入非线性权重卡尔曼估计并结合X射线变换部分数据结果来实现，还建立了新的卡尔曼估计。
中文摘要 AI 辅助
本文致力于抛物型方程中依赖于时间和空间变量的零阶和一阶系数的反问题，通过狄利克雷激励产生的通量的部分边界测量来确定。具体而言，我们从与相应抛物型方程相关的部分狄利克雷 - 诺伊曼映射中确定与时间相关的对流项和势，其中诺伊曼测量限于从位于区域闭包外一点照亮的边界部分。主要目标是将迄今限于椭圆型情形的若干观察结果扩展到抛物型方程，并利用这些性质恢复依赖于时间和空间变量的一般系数类。为此，我们为抛物型方程开发了一类合适的特殊解，引入了具有非线性权重的新的卡尔曼估计族，并将这些工具与X射线变换的部分数据结果相结合。当前工作的主要困难在于，与抛物型方程部分数据反问题的现有文献不同，特殊解的相位和相应卡尔曼估计的权重都是非线性的。作为分析的副产品，我们建立了一类具有非线性权重的抛物型方程的新卡尔曼估计，可视为椭圆型情形中极限卡尔曼权重概念的抛物型类似物。
英文摘要
This article is devoted to the inverse problem of determining the zeroth- and first-order coefficients, depending on both the time and space variables, in a parabolic equation from partial boundary measurements of the flux generated by Dirichlet excitations. More precisely, we establish the unique determination of a time-dependent convection term and potential from the partial Dirichlet-to-Neumann map associated with the corresponding parabolic equation, where the Neumann measurements are restricted to the portion of the boundary illuminated from a point located outside the closure of the domain. Our main objective is to extend to parabolic equations several observations that have thus far been confined to the elliptic setting and to exploit these properties to recover a general class of coefficients depending on both time and space variables. To achieve this, we develop a suitable class of special solutions to the parabolic equation, introduce a new family of Carleman estimates with nonlinear weights, and combine these tools with partial data results for the X-ray transform. The principal difficulty of the present work stems from the fact that, unlike the existing literature on partial data inverse problems for parabolic equations, both the phase of the special solutions and the weight of the corresponding Carleman estimates are nonlinear. As a byproduct of our analysis, we establish a new class of Carleman estimates for parabolic equations with nonlinear weights, which may be viewed as the parabolic analogue of the notion of limiting Carleman weights in the elliptic setting.
