Conservation laws and Hamilton–Jacobi equations with space inhomogeneity

Abstract Conservation laws with an x -dependent flux and Hamilton–Jacobi equations Hamiltonian are considered within the same set of assumptions. Uniqueness stability estimates obtained only requiring sufficient smoothness flux/Hamiltonian. Existence is proved without any convexity assumptions under a mild coercivity hypothesis. The correspondence between semigroups generated by these fully detailed. With respect to classical Kružkov approach conservation laws, we relax definition solution avoid restriction on growth flux. A key role played construction sufficiently many entropy stationary solutions in $${{\textbf{L}}^\infty }$$ L ∞ that provide global bounds time space.