AI 中文总结
研究具有不连续通量的多维标量守恒律的适定性，通过特殊方法处理通量非平凡线性组合恒定时的障碍，构建理论并扩展到特定数据类型，建立了存在性、唯一性和局部\(L^1\)稳定性。
AI 中文摘要
我们建立了具有不连续异质通量的多维标量守恒律的存在性、唯一性和局部\(L^1\)稳定性，而不对物理时空通量施加非退化条件。不连续集可以是一族局部有限的\(C^2\)超曲面，其交集和奇点包含在一个\((d - 1)\)维豪斯多夫测度为零的闭集中。主要障碍是当时空通量分量的非平凡线性组合在一个状态区间上恒定时，多维紧致性可能失效。对于界面问题，这种障碍可归结为物理法向通量。其平坦区间通过保持法向通量从而保持兰金 - 于戈尼奥关系的法向通量商来压缩。在非平坦区域，时间和切向分量由法向通量的辅助多项式函数代替，而物理法向分量不变。由此产生的辅助时空向量满足帕诺夫紧致性理论所需的区间非退化性。这产生了商变量的强紧致性和强单侧迹。界面可容许性条件通过投影与两个单侧物理法向通量相关的零粘性胚得到。局部粘性比较表明商迹属于这个最大\(L^1\)耗散胚，并且一个杨测度收缩论证恢复了物理状态的强收敛。弯曲界面通过扁平化、局部化和有限传播修补来处理。该理论首先针对\(BV\)初始数据构建，然后通过连续性扩展到取值于不变区间的\(L^1_{\mathrm{loc}}\cap L^\infty\)数据。
英文摘要
We establish existence, uniqueness, and local \(L^1\)-stability for multidimensional scalar conservation laws with discontinuous heterogeneous flux, without imposing a non-degeneracy condition on the physical space--time flux. The discontinuity set may be a locally finite family of \(C^2\) hypersurfaces whose intersections and singular points are contained in a closed set of vanishing \((d-1)\)-dimensional Hausdorff measure. The main obstacle is the possible failure of multidimensional compactness when a nontrivial linear combination of the space--time flux components is constant on an interval of states. We show that, for interface problems, this obstruction can be reduced to the physical normal flux. Its flat intervals are collapsed by a normal-flux quotient that preserves the normal flux and hence the Rankine--Hugoniot relation. On the non-flat region, the time and tangential components are replaced by auxiliary polynomial functions of the normal flux, while the physical normal component is left unchanged. The resulting auxiliary space--time vector satisfies the interval non-degeneracy required by Panov's compactness theory. This yields strong compactness and strong one-sided traces of the quotient variables. The interface admissibility condition is obtained by projecting the vanishing-viscosity germ associated with the two one-sided physical normal fluxes. A localized viscous comparison shows that the quotient traces belong to this maximal \(L^1\)-dissipative germ, and a Young-measure contraction argument recovers strong convergence of the physical states. Curved interfaces are treated by flattening, localization, and finite-propagation patching. The theory is first constructed for \(BV\) initial data and then extended by continuity to \(L^1_{\mathrm{loc}}\cap L^\infty\) data taking values in the invariant interval.
