标题： 非光滑速度场流的$θ$-方法的先验误差估计 显示英文标题

标题： A priori error estimates for the $θ$-method for the flow of nonsmooth velocity fields

摘要： 具有低正则性（低于Lipschitz阈值）的速度场自然出现在许多数学物理模型中，例如非齐次不可压缩Navier-Stokes方程，并在非线性偏微分方程的分析中起着根本作用。 DiPerna-Lions理论确保了与具有Sobolev正则性的散度自由速度场相关的流的存在性和唯一性。 本文建立了先验误差估计，表明通过$\theta$-方法构造的数值解以对数速率收敛到具有Sobolev正则性的速度场的精确（解析）流。 此外，我们还推导了与传输方程的拉格朗日解相关的类似先验误差估计，显示出相同的对数收敛率。 我们的理论结果由数值实验支持，这些实验验证了预测的对数行为。 显示英文摘要

摘要： Velocity fields with low regularity (below the Lipschitz threshold) naturally arise in many models from mathematical physics, such as the inhomogeneous incompressible Navier-Stokes equations, and play a fundamental role in the analysis of nonlinear PDEs. The DiPerna-Lions theory ensures existence and uniqueness of the flow associated with a divergence-free velocity field with Sobolev regularity. In this paper, we establish a priori error estimates showing a logarithmic rate of convergence of numerical solutions, constructed via the $\theta$-method, towards the exact (analytic) flow for a velocity field with Sobolev regularity. In addition, we derive analogous a priori error estimates for Lagrangian solutions of the associated transport equation, exhibiting the same logarithmic rate of convergence. Our theoretical results are supported by numerical experiments, which confirm the predicted logarithmic behavior.