Regularity for Hele-Shaw flow with sign-changing sources

Characterize the regularity of the free boundary for Hele-Shaw flow when the source term changes sign, where the correspondence between the constrained transport formulation and the pressure formulation breaks down.

Background

The paper studies Hele-Shaw flow with a nonnegative source term and develops regularity and space-time geometric results under positivity and boundedness assumptions on the source. The authors note that allowing the source to change sign destroys the correspondence between the density equation and the pressure formulation used throughout the analysis.

The regularity theory in this sign-changing regime is therefore identified as an unresolved issue, and it lies outside the scope of the paper’s methods.

References

On the other hand, when $c$ changes sign, the correspondence between `eqn:densityand \eq:intro-HS-law` breaks down altogether, and the regularity of the free boundary remains largely open .

eqn:density:

∂tρ−∇⋅(ρ∇p)=cρ,0≤ρ≤1.\partial_t\rho - \nabla\cdot (\rho \nabla p) = c\rho, \quad 0\leq \rho \leq 1.

— Space-time geometry of Hele-Shaw flow  (2608.16509 - Collins et al., 17 Aug 2026) in Section 1, subsection “Setting and standing assumptions”

Rather than carrying out proofs, we confine ourselves to indicating the parts of our analysis where modifications are necessary; the full verification remains open.

— Space-time geometry of Hele-Shaw flow  (2608.16509 - Collins et al., 17 Aug 2026) in Section 9, “Application to the injection problem”