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A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux
Boris Andreianov, Kenneth H. Karlsen, Nils Henrik Risebro
TL;DR
Discontinuous-flux conservation laws can have multiple L1-contractive semigroups, making admissibility and uniqueness dependent on the chosen physical selection. The paper encodes that selection through germs of elementary stationary solutions, develops trace and entropy formulations, and connects definite germs to unique contractive solvers while discussing convergence-based existence methods.
Problem
Discontinuous-flux conservation laws lack a comparable well-posedness theory, and the same equation may admit multiple L1-contractive semigroups associated with different physical phenomena.
Method
The paper selects a germ of elementary solutions and formulates admissibility through boundary traces and global entropy inequalities, with measure-valued and approximation-based convergence tools.
Results
Each maximal L1D germ yields an L1-contractive semigroup, while a definite germ corresponds to a unique L1-dissipative solver.
Takeaways & Limitations
Germs provide a unified way to compare admissibility criteria and to connect selected elementary solutions with uniqueness, contraction, and existence results.
Takeaways & Limitations
Uniqueness of G-entropy process solutions for a definite germ remains an open problem in general.
Abstract
from arXiv · showhide
We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$Γ$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.
1. Introduction
The paper studies why discontinuous-flux conservation laws admit multiple L1-contractive solution semigroups and proposes a germ-based framework for admissibility, uniqueness, and convergence.
- Motivation: Discontinuous-flux conservation laws lack a comparable well-posedness theory, while the same equation can admit multiple L1-contractive semigroups representing different physical phenomena.Weak solutions are generally nonunique, so additional admissibility criteria are required.
- Model: The framework focuses on one-dimensional conservation laws with continuous left and right fluxes that may be less regular than commonly assumed.The initial datum lies in L∞(R; U), and the source term can be incorporated in suitable cases.
- Germ-based admissibility: Admissibility is axiomatized by selecting a set G of elementary stationary solutions across the flux discontinuity.These elementary solutions encode the allowed pairs of left and right states at x=0.
- Uniqueness: For a definite germ G, the associated entropy formulation yields a unique L1-dissipative solver whose traces lie in the unique maximal L1D extension G*.The formulation uses boundary traces and global entropy inequalities, with G-entropy solutions providing the associated solution concept.
- Convergence: The selected elementary solutions are exactly the elementary solutions contained in the associated solver, and process-solution results support strong convergence of bounded approximate solutions.The convergence argument does not require BV-type a priori estimates when the stated existence and compatibility conditions hold.
2. Preliminaries
The preliminaries recall Kruzhkov entropy solutions and establish strong trace tools for continuous fluxes, including degenerate cases handled through singular mappings.
- Kruzhkov theory: Kruzhkov entropy solutions use |z-k| as entropy and an associated entropy flux, providing the local entropy framework for scalar conservation laws.The recalled definition is localized to open space-time domains.
- Strong traces: Entropy solutions admit strong initial traces, and analogous half-space trace results describe behavior at spatial boundaries.The half-space result requires the stated non-degeneracy condition on the flux.
- Degenerate fluxes: For continuous fluxes, a singular mapping V(u) admits a strong one-sided trace even when the flux may be constant on some intervals.The corresponding left and right trace statements apply on the two sides of the discontinuity.
- Trace representation: Fluxes and entropy fluxes can be represented continuously in the transformed variable V, allowing their boundary traces to be recovered from traces of V(u).When the flux is non-degenerate, V is the identity and the transformed entropy fluxes coincide with the usual ones.
3. The model one-dimensional problem
The model defines admissibility through germs of stationary elementary solutions at the flux discontinuity and establishes equivalent entropy formulations, uniqueness, contraction, comparison, and continuous dependence results under germ conditions.
- Model and germs: The flux is constant on each side of the discontinuity line Σ, where admissibility is specified through left and right entropy fluxes.The entropy fluxes q^l,r are represented using continuous functions and mappings V^l,r.
- Model and germs: A germ is a set of pairs satisfying Rankine–Hugoniot, and it is L1-dissipative when the pairwise boundary inequality holds for all its elements.For a definite germ, the dual germ is its unique maximal L1D extension; maximal L1D germs are self-dual.
- Entropy solutions: Pairs in a germ generate admissible piecewise constant stationary solutions, while the dual germ characterizes exactly which such profiles are G-entropy solutions.Thus the elementary solutions underlying admissibility are explicitly identified through G and G∗.
- Entropy solutions: For a definite germ, G-entropy solutions are unique, satisfy the Kato inequality and L1 contractivity, and generate a strongly L1-continuous semigroup.The associated semigroup is an L1-dissipative solver.
- Entropy formulations: The trace and global entropy-inequality definitions are equivalent for any L1D germ, and the model case can use zero-remainder inequalities for selected test states.The trace formulation also yields strong left and right traces for self-similar Riemann solutions.
- Stability and dependence: Definite germs provide comparison, maximum-principle, and L∞ estimates, while the germ distance yields continuous dependence on the choice of germ.The distance estimate includes an initial-data term and a time-scaled germ-distance term; zero distance makes maximal L1D germs coincide.
- Completeness and process solutions: Existence for all bounded initial data forces the dual germ to be complete, and maximal L1D germs correspond to L1-contractive semigroups under the stated invariance assumptions.A maximal L1D germ also admits a unique process solution whenever an entropy solution already exists, but uniqueness of process solutions is open in general.
4. Examples and analysis of known admissibility criteria
The paper shows how established admissibility criteria correspond to specific germs, with definiteness, completeness, and L1-dissipativity determining well-posedness and uniqueness properties. Examples include Kruzhkov, vanishing-viscosity, Audusse–Perthame, Rankine–Hugoniot, and Karlsen–Risebro–Towers germs.
- Germ structure: Complete L1D germs are maximal and definite, while definite germs yield unique maximal L1D extensions and associated entropy-solution theories.These structural properties organize when germ-based admissibility leads to well-posedness.
- Kruzhkov and vanishing viscosity: The Kruzhkov germ is complete and produces precisely the classical Kruzhkov entropy solutions, including existence and well-posedness for bounded data.Its associated maximal L1D germ coincides with the vanishing-viscosity germ.
- Kruzhkov and vanishing viscosity: The vanishing-viscosity germ is L1-dissipative by construction, and its closure is the unique maximal L1D extension of the germ generated by explicit viscosity limits.The closure is identified with the germ defined by the non-strict inequality version of the admissibility condition.
- Monotone fluxes: For monotone fluxes, singleton germs determined by Rankine–Hugoniot connections are definite and complete, yielding infinitely many non-equivalent L1-contractive semigroups when multiple flux levels match.The Rankine–Hugoniot germ itself can remain maximal but fail to be complete in the non-decreasing/non-increasing configuration.
- Audusse–Perthame criterion: The Audusse–Perthame germ is complete, and its entropy solutions coincide with those generated by the corresponding singleton connection germ.This identifies the adapted-entropy condition with a particular complete definite germ.
- Karlsen–Risebro–Towers criterion: The Karlsen–Risebro–Towers germ is maximal L1D under the crossing condition, whereas uniqueness may fail for general fluxes when that condition is violated.The condition is sufficient for L1-dissipativity but not necessary for uniqueness of the associated germ-entropy solutions.
5. The vanishing viscosity germ
The vanishing viscosity germ characterizes stationary interface states selected by vanishing-viscosity approximations and is identified with a maximal L1D germ under the paper’s assumptions.
- Construction: The standing-wave vanishing viscosity approach defines admissible pairs through stationary profiles connecting left and right states across the flux discontinuity.The profile’s monotonicity yields the required flux inequalities on either side of an intermediate state.
- Maximality: The standing-wave germ is definite and has the same unique maximal L1D extension as the explicitly described germ.The paper also states that the corresponding entropy solutions coincide.
- Characterization: The vanishing viscosity germ is explicitly described by alternatives involving equal states or one-sided flux inequalities relative to a common flux level.For ordered states, the conditions require inequalities on intervals split at an intermediate state.
- Terminology: The maximal L1D germ associated with vanishing viscosity is denoted GV V and is called the vanishing viscosity germ for f l,r.It is defined for general continuous left and right fluxes, although it need not be complete.
- Relation to prior conditions: Diehl’s Γ-condition coincides with the explicit vanishing-viscosity characterization and is related to Oleinik and chord conditions through a traveling-wave argument.The Γ-condition requires equal interface fluxes and an intermediate state satisfying the stated inequalities.
- Approximation: Under suitable approximation conditions, regularized stationary solutions converge locally to the elementary stationary solution selected by the germ.For the smoothed problem, stationary profiles exist when δ/ε ≤ 2/L, followed by convergence as ε decreases.
6. Some existence and convergence results
The paper establishes existence, uniqueness, contraction, and convergence results for entropy solutions associated with vanishing-viscosity and general maximal L1D germs under progressively broader assumptions.
- Approximate solutions: The framework also proves convergence of uniformly bounded approximate solutions without BV a priori estimates when the approximation is compatible with the germ.For merely continuous fluxes, existence additionally assumes that solutions exist for all bounded initial functions.
- Standard vanishing viscosity: For Lipschitz, non-affine fluxes on [0, 1] vanishing at 0 and 1, every measurable [0, 1]-valued datum has a unique GV V-entropy solution.The solution is obtained as a vanishing-viscosity limit, and the germ is definite and complete.
- Standard vanishing viscosity: If initial data converge almost everywhere, the viscous approximations converge almost everywhere to the unique GV V-entropy solution.The uniqueness of the accumulation point ensures convergence of the whole sequence.
- Standard vanishing viscosity: The standard approximation preserves the bounds 0 ≤ uε ≤ 1 and yields comparison, contraction, strong traces, and Kruzhkov entropy inequalities away from the interface.These properties support passage to the limit and identification of the limiting interface entropy condition.
- Adapted viscosity: For a prescribed connection (A, B), an adapted viscosity constructs the unique G(A,B)-entropy solution and converges almost everywhere to it.The method encodes the selected connection through the viscosity and recovers the corresponding global entropy inequality.
- General germs: For any complete maximal L1D germ and locally Lipschitz fluxes on R, every bounded initial datum admits a unique G-entropy solution.The construction uses contraction and approximation arguments beyond the specific vanishing-viscosity germ.
- Limitations: The adapted-viscosity convergence theory has a restrictive assumption: in general, (B2) is too strong and is difficult to justify for some schemes.The adapted approximation preserves only the connection solution and the constant states 0 and 1 in the cited example.