Prove the unestablished properties of the multidimensional generalized tangent-vector calculus

Establish, for the multidimensional generalized tangent-vector approach to scalar hyperbolic balance laws, the several properties that have been proved for the one-dimensional generalized tangent-vector theory but remain unproved in the multidimensional setting.

Background

The paper extends generalized tangent-vector ideas from one-dimensional balance laws to multidimensional scalar balance laws with shock hypersurfaces. In the one-dimensional theory, several structural and analytical properties of generalized tangent vectors have already been established. The multidimensional construction introduced here consists of an $L^1$ variation together with a normal displacement of the shock hypersurface, but the paper explicitly notes that the corresponding properties have not yet been proved for this higher-dimensional approach.

This unresolved issue concerns the mathematical foundation of the proposed sensitivity calculus: beyond deriving formal evolution equations and presenting numerical evidence, the multidimensional generalized tangent-vector framework requires rigorous proofs of the analogous properties known in one dimension. The paper does not specify the individual properties in the cited passage.