Regularity for a singular conservation law

AbstractThe main structure underlying the nonlinearity of conservation laws of gasdynamical type in two independent variables will be discussed at the hand of a canonical example describing also properties of water waves near shore. The ultimately singular nature of such laws is here the central issue and calls for an unusual formulation (Sect. 2). Attention is directed to the globally strong solutions, and an unusual regularization (Sect. 2) is employed to make them accessible, after illposedness is overcome (Sect. 3). The usual regularity theory is not normally sufficient for singular partial differential equations, and the necessary additional chapter on extensions to the singular locus is developed (Sect. 4) in detail for the canonical example. Criteria for the relation between regularized and strong solutions are discussed in Section 5 and used in Section 6 to characterize the class of solutions that are globally strong in the strictest sense