GRAPH SOLUTIONS OF NONLINEAR HYPERBOLIC SYSTEMS

For nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) we introduce a geometric framework in which solutions are sought as (continuous) parametrized graphs(t,s) ↦ (X,U)(t,s) satisfying ∂sX ≥ 0, rather than (discontinuous) functions (t,x) ↦ u(t,x). On one hand, we generalize an idea by Dal Maso, LeFloch, and Murat who used a family of traveling wave profiles to define non-conservative products, and we define the notion of graph solution subordinate to a family of Riemann graphs. The latter naturally encodes the graph of the solution to the Riemann problem, which should be determined from an augmented model taking into account small-scale physics and providing an internal structure to the shock waves. In a second definition, we write an evolution equation on the graphs directly and we introduce the notion of graph solution subordinate to a diffusion matrix, which merges together the hyperbolic equations (in the "non-vertica...

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