The Legacy of Olga Oleǐnik in Hyperbolic Conservation Laws

Systems of conservation laws are partial differential equations that model the dynamics of continua — fluids, plasmas or elastic solids — by means of the fundamental principles of conservation of mass, momentum, and energy, supplemented by constitutive relations, typically based in thermodynamics or other classical physics. The standard form of a conservation law system, in one space dimension, is Ut + F (U)x = 0, where U ∈ R is a vector of states, or conserved quantities, and F is a (linear or nonlinear) flux function. The familiar wave equation, (ρut)t = (Tux)x, for the vertical displacement u of a stretched string of constant linear density ρ and constant tension T , expresses conservation of momentum and assumes the standard form if we define c2 = T/ρ, and