**Francesca Marcellini**

**Abstract:**
In a class of systems of balance laws
in several space dimensions,
we prove the stability of solutions
with respect to variations in the flow and in the source.
This class comprises a model
describing the cutting of metal plates
by means of laser beam is proved to admit solutions.

**Francesca Marcellini**

**Abstract:**
We consider the initial boundary value problem
for the phase transition traffic model introduced in
[Colombo–Marcellini–Rascle 2010],
which is a macroscopic model based on a 2×2
system of conservation laws.
We prove existence of solutions by means of
the wave-front tracking technique,
provided the initial data and the boundary conditions
have finite total variation.

Starting January 1, 2017, we will no longer accept new submissions. Authors will still be welcome to submit revisions to papers already published. The contents will remain available for the foreseeable future. It has been a good 21 years, but it is time to admit that most people prefer to put their papers on arXiv instead. We urge you to do the same.

**Marin Prebeg, Tore Flåtten and Bernhard Müller**

**Abstract:**
We present Large Time Step (LTS) extensions
of the Harten–Lax–van Leer (HLL)
and Harten–Lax–van Leer Contact (HLLC) schemes.
Herein, LTS denotes a class of explicit methods
stable for Courant numbers greater than one.
The original LTS method
[Randall J. LeVeque:
A large time step generalization of Godunov's method
for systems of conservation laws.
*SIAM J. Numer. Anal.* **22** (1985), no. 6,
1051–1073]
was constructed as an extension of the Godunov scheme,
and successive versions have been developed
in the framework of Roe's approximate Riemann solver.

We first formulate the LTS extension of the original HLL scheme in conservation form. Next, we provide explicit expressions for the flux-difference splitting coefficients and the numerical viscosity coefficients. We then formulate the LTS extension of the HLLC scheme in conservation form.

We apply the new schemes to the one dimensional Euler equations and compare them to their non-LTS counterparts. As test cases, we consider the classical Sod shock tube problem and the Woodward–Colella blast-wave problem. It is shown that the LTS–HLL scheme smears out the contact discontinuity, while the LTS–HLLC scheme improves the resolution of both shocks and contact discontinuities. In addition, we numerically demonstrate that for the right choice of wave velocity estimates both schemes calculate entropy satisfying solutions.

**Edda Dal Santo, Massimiliano D. Rosini, Nikodem Dymski, and Mohamed Benyahia**

**Abstract:**
We generalize the phase transition model studied in
[R. Colombo. Hyperbolic phase transition in traffic flow.
*SIAM J. Appl. Math.*, **63**(2):708–721, 2002],
that describes the evolution of vehicular traffic along a one-lane road.
Two different phases are taken into account, according to whether the traffic is low or heavy.
The model is given by a scalar conservation law
in the *free-flow* phase
and by a system of two conservation laws
in the *congested* phase.
In particular, we study the resulting Riemann problems
in the case a local point constraint on the flux of the solutions is enforced.

**Laura Caravenna and Gianluca Crippa**

☞ External link to arXiv

**Abstract:**
We deal with the uniqueness of distributional solutions
to the continuity equation with a Sobolev vector field
and with the property of being a Lagrangian solution,
i.e. transported by a flow of the associated ordinary differential equation.
We work in a framework of low local integrability of the solution,
in which the classical DiPerna–Lions theory of uniqueness
and Lagrangianity of distributional solutions
does not apply due to the insufficient integrability of the commutator.
We introduce a general principle to prove that a solution is Lagrangian:
we rely on a disintegration along the unique flow
and on a new directional Lipschitz extension lemma,
used to construct a large class of test functions
in the Lagrangian distributional formulation of the continuity equation.

**Kenneth H. Karlsen and Erlend B. Storrøsten**

*Updated* 2016-08-22

**Abstract:**
We analyze a semi-discrete splitting method for
conservation laws driven by a semilinear noise term.
Making use of fractional $BV$ estimates, we
show that the splitting method produces a
compact sequence of approximate solutions converging
to the exact solution, as the time step $\Delta t \to 0$.
Under the assumption of a homogenous noise function, and thus
the availability of $BV$ estimates, we provide an $L^1$ error estimate.
Bringing into play a generalization of
Kružkov's entropy condition, permitting
the “Kružkov constants” to be Malliavin
differentiable random variables, we establish an
$L^1$ convergence rate of order $\frac13$ in $\Delta t$.

**Stefano Bianchini and Elio Marconi**

**Abstract:**
We prove that if $u$ is the entropy solution
to a scalar conservation law in one space dimension,
then the entropy dissipation is a measure concentrated
on countably many Lipschitz curves.
This result is a consequence of a detailed analysis
of the structure of the characteristics.

In particular the characteristic curves
are segments outside a countably 1-rectifiable set,
and the left and right traces of the solution exist
in a $C^0$-sense up to the degeneracy
due to the segments where $f''=0$.
We prove also that the initial data
is taken in a suitably strong sense,
and we give some counterexamples
which show that these results are sharp.

**Mauro Garavello and Francesca Marcellini**

**Abstract:**
We extend the Phase Transition model for traffic
proposed in [7], by Colombo, Marcellini, and Rascle
to the network case.
More precisely, we consider the Riemann problem
for such a system at a general junction with
*n* incoming and *m* outgoing roads.
We propose a Riemann solver at the junction
which conserves both the number of cars
and the maximal speed of each vehicle,
which is a key feature of the Phase Transition model.
For special junctions,
we prove that the Riemann solver is well defined.

**Rinaldo M. Colombo and Mauro Garavello**

*Updated* and title revised 2015-07-07
– updated again 2016-05-10

**Abstract:**
A biological resource is a population characterized by
birth, aging and death, grown in order to produce a profit.
The evolution of this system is described by
a structured population model,
modified to take into account the selection for reproduction
or for the market.
This selection is the control that has to be
optimized in order to maximize the profit.
First we prove the well posedness of the descriptive model.
Then, the profit is shown to be Gâteaux differentiable
with respect to the controls.
Finally, we ensure that the maximal profit
can be reached by means of Bang–Bang controls.

**Mauro Garavello and Stefano Villa**

**Abstract:**
We study the Cauchy problem for the
Aw–Rascle–Zhang model for traffic flow
with a flux constraint at $x=0$.
More precisely we consider the Riemann solver,
conserving the number of cars at $x=0$
but not the generalized momentum,
introduced in [9] for the problem with flux constrained.
For such a Riemann solver,
we prove existence of a solution for the Cauchy problem.
The proof is based on the wave-front tracking method.
For the other Riemann solver in [9], existence of solution
to the Cauchy problem was proved in [1].

**John D. Towers**

**Abstract:**
We consider the Godunov scheme as applied to a scalar conservation law
whose flux has discontinuities in both space and time.
The time and space dependence of the
flux occurs through a positive multiplicative coefficient.
That coefficient has a spatial
discontinuity along a fixed interface at $x=0$.
Time discontinuities occur in the coefficient
independently on either side of the interface.
This setup applies to the LWR traffic model
in the case where different time-varying speed limits are imposed
on different segments of a road.
We prove that approximate solutions
produced by the Godunov scheme converge to the unique entropy solution,
as defined in G.M. Coclite and N.H. Risebro,
*Conservation Laws with time dependent discontinuous coefficients*,
SIAM J. Math. Anal., 36 (2005).
Convergence of the Godunov scheme
in the presence of spatial flux discontinuities
alone is a well established fact.
The novel aspect of this paper is convergence in the presence of
additional temporal flux discontinuities.

**Mohamed Benyahia and Massimiliano D. Rosini**

**Abstract:**
In this paper, we consider the two phases macroscopic traffic model
introduced in
[P. Goatin, The Aw-Rascle vehicular traffic flow with phase transitions,
*Mathematical and Computer Modeling* **44** (2006) 287–303].
We first apply the wave-front tracking method to prove existence
and a priori bounds for weak solutions.
Then, in the case the characteristic field
corresponding to the free phase is linearly degenerate,
we prove that the obtained weak solutions are in fact entropy solutions
*à la* Kruzhkov.
The case of solutions attaining values at the vacuum is considered.
We also present an explicit numerical example
to describe some qualitative features of the solutions.

**François Bouchut and Xavier Lhébrard**

**Abstract:**
The shallow water magnetohydrodynamic system
involves several families of physically relevant steady states.
In this paper we design a well-balanced numerical scheme
for the shallow water magnetohydrodynamic system with topography,
that resolves exactly a large range of steady states.
Two variants are proposed with slightly different
families of preserved steady states.
They are obtained by a generalized
hydrostatic reconstruction algorithm
involving the magnetic field
and with a cutoff parameter to remove singularities.
The solver is positive in height and semi-discrete entropy satisfying,
which ensures the robustness of the method.

**François Bouchut and Xavier Lhébrard**

**Abstract:**
The shallow water magnetohydrodynamic system
describes the thin layer evolution of the solar tachocline.
It is obtained from the three dimensional incompressible
magnetohydrodynamic system
similarly as the classical shallow water system
is obtained from the incompressible Navier–Stokes equations.
The system is hyperbolic and has two additional waves
with respect to the shallow water system, the Alfven waves.
These are linearly degenerate, and thus do not generate dissipation.
In the present work we introduce a 5-wave approximate Riemann solver
for the shallow water magnetohydrodynamic system,
that has the property to be non dissipative on Alfven waves.
It is obtained by solving a relaxation system of Suliciu type,
and is similar to HLLC type solvers.
The solver is positive and entropy satisfying, ensuring its robustness.
It has sharp wave speeds, and does not involve any iterative procedure.

**Alberto Bressan and Tianyou Zhang**

**Abstract:**
The paper is concerned with the Burgers–Hilbert equation
$u_t + (u^2/2)_x = \mathbf{H}[u]$,
where the right hand side is a Hilbert transform.
Unique entropy admissible solutions are constructed,
locally in time, having a single shock.
In a neighborhood of the shock curve,
a detailed description of the solution is provided.