We consider divergence form, second-order strongly parabolic systems in a cylindrical domain with a finite number of subdomains under the assumption that the interfacial boundaries are \(C^{1,\text {Dini}}\) and \(C^{\gamma _{0}}\) in the spatial variables and the time variable, respectively. Gradient estimates and piecewise \(C^{1/2,1}\)-regularity are established when the leading coefficients and data are assumed to be of piecewise Dini mean oscillation or piecewise Hölder continuous. Our results improve the previous results in Fan et al. (Electron J Differ Equ 2013:1–24, 2013) and Li and Li (Sci China Math 60(11):2011–2052, 2017) to a large extent, and appear to be the first of its kind for time-dependent subdomains. As a byproduct, we obtain optimal regularity of weak solutions to parabolic transmission problems with \(C^{1,\mu }\) or \(C^{1,\text {Dini}}\) interfaces. This gives an extension of a recent result in Caffarelli et al. (Regularity for \(C^{1,\alpha }\) interface transmission problems. arXiv:2004.07322 [math.AP]) to parabolic systems.

在界面边界为\（C ^ {1，\ text {Dini}} \）和\（C ^ { \ gamma _ {0}} \）分别在空间变量和时间变量中。梯度估计和分段\（C ^ {1 / 2,1} \）当假定前导系数和数据为分段迪尼平均振动或分段Hölder连续时，将建立-正则性。我们的结果改进了Fan等人以前的结果。（Electron J Differ Equ 2013：1–24，2013）和Li and Li（Sci China Math 60（11）：2011–2052，2017）在很大程度上，并且似乎是时间依赖的同类中的第一个子域。作为副产品，我们使用\（C ^ {1，\ mu} \）或\（C ^ {1，\ text {Dini}} \）接口获得抛物线传递问题的弱解的最佳正则性。这扩展了Caffarelli等人的最新结果。（\（C ^ {1，\ alpha} \）的规则性接口传输问题。arXiv：2004.07322 [math.AP]）到抛物线系统。