Existence theory for nonlinear hyperbolic systems in nonconservative form

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

“…In effect, adding to (1) the equation ∂H ∂t = 0, the system can be rewritten in this form (see [12], [6], [7], [9], [10]). The nonconservative products involved in (2) do not make sense in general within the framework of distributions.…”

“…In practical applications, it has to be based on the physical background of the problem. In [12] a clear motivation for the selection of the family of paths is provided when a physical regularization by diffusion, dispersion, etc is available. Nevertheless, it is natural from the mathematical point of view to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields.…”

The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section

“…To deal with discontinuous cross section, following [17] we supplement (1.1) with the "trivial" equation 2) so that the whole set of equations can be written as a hyperbolic system in nonconservative form…”

“…In consequence, at least within the regime where the system is strictly hyperbolic, the theory of such systems developed by LeFloch and co-authors (see [7] and also [16][17][18][19][20][21][22]) applies, and provide the existence of entropy solutions to the Riemann problem (a single discontinuity separating two constant states as an initial data), as well as to the Cauchy problem (for solution with sufficiently small total variation). More recently, LeFloch and Thanh [21,22] solved the Riemann problem for arbitrary data, including the regime where the system fails to be globally strict hyperbolicity (i.e., the resonant case).…”

Uniqueness for Multidimensional Hyperbolic Systems with Commuting Jacobians

“…Bressan and LeFloch [8] proved that the Cauchy problem for (1.1) (with d = 1) admits at most one entropy solution satisfying the tame variation condition, which requires, in essence, that the total variation on an interval at a given time controls the total variation along any space-like curve included in the domain of determinacy of the given interval. The tame variation property is satisfied by solutions constructed, for instance, by the Glimm scheme ( [27,25,20,21,28,3] for recent works) or by the vanishing viscosity method [4], and, therefore, the theorem in [8] provides a uniqueness result in the same class where the existence is known. Later, it was observed [7,9] that the uniqueness result remains true under even weaker conditions (tame oscillation or bounded variation on spacelike lines).…”