Well-Posedness of the Cauchy Problem for First-order Quasilinear Equations with Non-Lipschitz Source Terms and Its Applications

The paper’s Theorem 2.4 constructs multidimensional Riemann solutions for ut + div f(u) = g(u) with two constant states across a smooth hypersurface using: (i) a shock that translates the initial interface by [χ](t) where [χ]' = [f]/[u], and (ii) a rarefaction determined by the implicit relation F(t,x,c)=M(x−χ(t,c))=0 and u(t,x)=ū(t,c) with ū solving u' = g(u). It verifies Rankine–Hugoniot and geometric entropy conditions in the shock case and proves Fc<0 in the rarefaction fan via Condition (H): Mxi f''i(u)>0, together with ūs≥0, ensuring uniqueness of c and that the PDE holds by implicit differentiation. These are exactly the constructions and checks performed in the candidate solution, down to the same definitions of ū, χ, [χ], and the same entropy/convexity mechanism. No substantive discrepancy was found; the model’s write-up mirrors Lemmas 6.2–6.3 and Theorem 2.4 of the paper and is correct under the paper’s hypotheses .