# Preprint 2010-007

# A theory of *L*^{1}-dissipative solvers for scalar conservation laws with discontinuous flux

## Boris Andreianov, Kenneth H. Karlsen and Nils Henrik Risebro

**Abstract:**
We propose a general framework for the study
of *L*^{1} contractive semigroups
of solutions to conservation laws
with discontinuous flux:

u_{t}+f(x,u)_{x}=0,
f(x,u)=f^{l}(u), x<0,
f(x,u)=f^{r}(u), x>0, (CL)

where the fluxes
f^{l}, f^{r}
are mainly assumed to be continuous.
Developing the ideas of a number of preceding works
(Baiti and Jenssen [14], Audusse and Perthame [12],
Garavello et al. [35], Adimurthi et al. [3],
Bürger et al. [21]),
we claim that the whole admissibility issue
is reduced to the selection of a family
of “elementary solutions”,
which are piecewise constant weak solutions of the form

c(x)=c^{l}**1**_{{x<0}}+c^{r}**1**_{{x>0}}.

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 “vanishing viscosity” germ,
which is a way to express the “Γ-condition” of
Diehl [32].
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 [80])
and in the form of a family of global entropy inequalities
(following Kružhkov [50] and Carrillo [22]).
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.

[3] Adimurthi, S. Mishra, and G. D. Veerappa Gowda. Optimal entropy solutions for conservation laws with discontinuous ﬂux-functions. J. Hyperbolic Differ. Equ., 2(4):783–837, 2005.

[12] E. Audusse and B. Perthame. Uniqueness for scalar conservation laws with discontinuous ﬂux via adapted entropies. Proc. Roy. Soc. Edinburgh A, 135(2):253–265, 2005.

[14] P. Baiti and H. K. Jenssen.
Well-posedness for a class of 2×2 conservation laws
with L^{∞} data.
J. Differential Equations, 140(1):161–185, 1997.

[21] R. Bürger, K. H. Karlsen, and J. Towers. An Engquist–Osher type scheme for conservation laws with discontinuous ﬂux adapted to ﬂux connections. SIAM J. Numer. Anal., 47:1684–1712, 2009.

[22] J. Carrillo. Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal., 147(4):269–361, 1999.

[32] S. Diehl. A uniqueness condition for non-linear convection–diffusion equations with discontinuous coefficients. J. Hyperbolic Differ. Equ., 6(1):127–159, 2009.

[35] M. Garavello, R. Natalini, B. Piccoli, and A. Terracina. Conservation laws with discontinuous ﬂux. Netw. Heterog. Media, 2:159–179, 2007.

[50] S. N. Kružkov. First order quasi-linear equations in several independent variables. Math. USSR Sbornik, 10(2):217–243, 1970.

[80] A. I. Vol'pert. The spaces BV and quasi-linear equations. Math. USSR Sbornik, 2(2):225–267, 1967.