Minimal assumptions to justify change-of-variable in the up-crossing entropy proof

Determine minimal regularity and integrability conditions on the initial datum u0: R -> R (with fixed c > 0 and 0 < ρ < 1) that both (i) ensure the auxiliary functions y(·), t(·), and x_t defined via the integral mappings I_L(y)=∫_y^{x0}(c−v0(z)) dz and I_R(x)=∫_{x0}^x(w0(z)−c) dz (with u0(x)=v0(x) for x ≤ x0 and u0(x)=w0(x) for x > x0, v0 < c < w0) are well-defined and possess the stated monotonicity properties, and (ii) rigorously justify the change of variable z = y(x) used to establish the Kruzhkov entropy inequality for the up-crossing configuration without assuming continuity of u0.

Background

In the up-crossing case (u0 below c for x ≤ x0 and above c for x > x0), the solution is constructed using auxiliary integral maps I_L and I_R and the function y = I_L{-1} ∘ I_R, along with t and its inverse x_t. Under continuity of u0, these objects are well-defined and the proof of the entropy inequality involves a change of variables z = y(x).

The authors argue that continuity of u0 is stronger than necessary. They observe that assuming u0 is locally integrable and that the set {u0 = c} has Lebesgue measure zero suffices to define y(·), t(·), and x_t with the desired properties. However, under this weaker assumption they could not rigorously justify the change-of-variable step required in the entropy proof.

The open question is to identify the minimal hypotheses on u0 that both preserve the construction of the auxiliary functions and fully justify the change-of-variable z = y(x) used in the Kruzhkov entropy argument for the up-crossing configuration.

References

As an example, assuming that $u_0$ is locally integrable and that $\mathcal{L}1\left({u_0=c}\right)=0$ (with $\mathcal{L}1$ being the Lebesgue measure on $R$) allows for a well-defined construction of the functions $y(\cdot),\;t(\cdot)$ and $x_\cdot$ together with all the desired properties. However, under this assumption, we were not able to rigorously justify the change of variable in the following section. A deeper analysis would be required to determine the minimal set of conditions under which the change of variable remains valid.

— Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context  (2604.01976 - Franke et al., 2 Apr 2026) in Subsubsection “On the regularity and integrability assumptions for u0” within Section “Second Type of Initial Condition: Up crossing”