Well-posedness and BV regularity for conservation laws with BV spatial flux in one and several space dimensions

Abstract: Well-posedness for conservation laws with discontinuous flux which contains flat regions in the state variable was unknown for quite some time. In this context, we propose a notion of entropy solution, which does not require the existence of traces and prove the uniqueness of the entropy solution. The existence of the entropy solution is established via a convergent Godunov type scheme. We also establish the convergence of front tracking approximations for the specific case of convex flux. The novelty of this work lies in the fact that no monotonicity or strict convexity of the flux in the state variable is assumed, resulting in the emergence of rarefactions at the interface and consequent increase in the number of fronts. In addition, a sufficient condition on the initial data and flux is coined to ensure a uniform BV bound on the entropy solutions. Counterexamples are constructed to exhibit the optimality of our assumptions, by showing the BV blow-up of the solution, not adhering to these assumptions.

We further prove the convergence of a Godunov type finite volume scheme to the entropy solution for a more general class of fluxes (Panov-type) in one as well as in several space dimensions. Moreover, our method of proof provides a spatial variation bound on the entropy solutions. Furthermore, we prove that our scheme converges to the unique entropy solution at an optimal rate of 1/2. Convergence of the Godunov type methods in multi-dimension and error estimates of the numerical scheme in one as well as in several dimensions are the first of its kind for conservation laws with discontinuous flux. Throughout this study we allow the flux to have infinitely many spatial dicontinuities which in turn may accumulate.

Lastly, we discuss the notion of generalized weak solution for a class of conservation laws with discontinuous flux which does not admit bounded weak solution. We prove the uniqueness and establish the existence via explicit formula as well as finite volume approximations. As an application, we use these tools to propose a Godunov type positivity preserving and entropy stable schemes for 2 × 2 systems of conservation laws that admit δ-shock solutions.