[Paper Review] Well-posedness for conservation laws with spatial heterogeneities and a study of BV regularity

Shyam Sundar Ghoshal, John D. Towers+1 more|arXiv (Cornell University)|Oct 26, 2020

Navier-Stokes equation solutions27 references4 citations

TL;DR

This paper establishes existence, uniqueness, and BV regularity for scalar conservation laws with spatially discontinuous and degenerate fluxes—where the flux may be flat over intervals—using a front-tracking algorithm and a novel entropy condition. It proves a BV bound under a new condition on initial data and constructs counterexamples showing the sharpness of this condition, resolving open problems in degenerate flux settings.

ABSTRACT

In this article, we consider scalar conservation laws with fluxes having spatial discontinuities and possible flat regions and study the following three aspects: (i) existence, (ii) uniqueness and (iii) BV regularity of solutions. We propose a uniqueness condition and prove existence of a weak solution via the method of wave front tracking. In the later part of the article, a BV bound of the solution is achieved under a suitable condition on the initial data and flux. We construct two counterexamples showing BV blow-up of the solution which proves the optimality on the assumptions

Motivation & Objective

• Address the open problem of uniqueness for scalar conservation laws with degenerate fluxes, where the flux has flat regions due to non-strictly convex or non-injective behavior.
• Establish existence of adapted entropy solutions for general discontinuous and degenerate fluxes via a front-tracking algorithm, filling a gap in the literature where such existence was previously unproven for this class.
• Provide a sufficient condition for bounded variation (BV) regularity of the entropy solution in the presence of infinitely many spatial discontinuities, a setting where BV blow-up is otherwise possible.
• Demonstrate optimality of the BV condition by constructing two counterexamples showing that BV blow-up occurs when the condition is violated.
• Extend the theory beyond the classical case of finitely many discontinuities to the more complex case of infinitely many discontinuities with accumulation points, particularly in degenerate flux regimes.

Proposed method

• Propose a new entropy condition tailored for degenerate fluxes (where the minimum of $ A(x,u) $ occurs on an interval $[a,b]$) to ensure uniqueness without requiring injectivity of the flux map.
• Use the front-tracking method to construct approximate solutions by solving Riemann problems at discontinuities and tracking wave interactions, proving convergence to a weak solution.
• Introduce a singular map technique to handle fluxes with flat regions and ensure stability and convergence of the front-tracking scheme.
• Establish compactness via total variation estimates by introducing a new condition on the initial data that controls the growth of variation across discontinuities.
• Construct two counterexamples where BV regularity fails under relaxed assumptions, proving the necessity and optimality of the proposed BV condition.
• Apply a Godunov-type approximation framework combined with measured-valued solutions to support existence and convergence results in the degenerate setting.

Experimental results

Research questions

• RQ1 Can a unique adapted entropy solution be guaranteed for scalar conservation laws with fluxes that are spatially discontinuous and degenerate (i.e., flat in $ u $) across intervals?
• RQ2 Is it possible to prove existence of a weak solution via front tracking for such degenerate and discontinuous fluxes, where previous methods failed?
• RQ3 What conditions on the initial data ensure that the total variation of the solution remains bounded in time, even with infinitely many spatial discontinuities?
• RQ4 Are the conditions for BV regularity optimal, or can BV blow-up occur under weaker assumptions?
• RQ5 How does the behavior of solutions differ in the case of infinitely many discontinuities compared to the classical case with finitely many discontinuities?

Key findings

• A new entropy condition is introduced that ensures uniqueness of the adapted entropy solution even when the flux $ A(x,u) $ is degenerate, i.e., achieves its minimum on an interval $[a,b]$ with $ a < b $.
• Existence of an adapted entropy solution is proven via front tracking for general discontinuous and degenerate fluxes, resolving an open problem in the literature.
• A sufficient condition on the initial data is identified under which the total variation of the solution remains uniformly bounded in time, even with infinitely many spatial discontinuities.
• Two counterexamples are constructed showing that BV blow-up occurs when the initial data condition is violated, proving that the BV bound condition is sharp and optimal.
• The solution exhibits infinite total variation at time $ t=1 $ in the counterexample, demonstrating that BV regularity cannot be expected without the proposed condition.
• The result shows a fundamental difference in regularity behavior between finitely and infinitely many discontinuities in the flux, particularly in degenerate cases.