VANISHING VISCOSITY ON A STAR-SHAPED GRAPH UNDER GENERAL TRANSMISSION CONDITIONS AT THE NODE

. In this paper we consider a family of scalar conservation laws deﬁned on an oriented star shaped graph and we study their vanishing viscosity approximations subject to general matching conditions at the node. In particular, we prove the existence of converging subsequence and we show that the limit is a weak solution of the original problem.

1. Introduction. We consider a family of scalar conservation laws defined on an oriented graph Γ consisting of m incoming and n outgoing edges Ω , = 1, . . . m + n joining at a single vertex. Incoming edges are parametrized by x ∈ (−∞, 0] while outgoing edges by x ∈ [0, ∞) in such a way that the junction is always located at x = 0. We use the index i, i = 1, . . . , m, to refer to incoming edges and j, j = m + 1, . . . , m + n, for the outgoing ones.
Finally, we introduce the necessary conservation assumption at the node, which transforms our family of independent equations into a single problem m i=1 f i (ρ i (t, 0−)) = m+n j=m+1 f j (ρ j (t, 0+)) for a.e. t ≥ 0.
Questions related to existence, uniqueness and stability of solutions for problems of this kind have been extensively investigated in recent years, mainly in relation with traffic modeling. The interested reader can refer to [7,13] for an overview of the subject. Here our point of view is different, as we do not focus on a specific model. We consider a parabolic regularization of the problem, similarly to what has been done in [11,10], but instead of enforcing a continuity condition at the node for the regularized solutions, we introduce a more general set of transmission conditions on the parabolic fluxes.
In this paper we modify the transmission condition of (4) and inspired by [14] we consider the following viscous approximation of (1), (2), and (3) where, of course, The additional assumptions we make on the functions β and on the initial conditions ρ ,0,ε are postposed to the next section.
The main result of the paper is the following.
It worth mentioning that a complete characterization of the limit solution obtained from (4) as ε → 0 is given in [3], where the authors adapt to a star shaped graph setting some ideas and techniques originally developed for conservation laws with discontinous flux, see in particular [2,4,5].
At the moment we are not able to formulate a similar characterization of the limit of (5). In general, however, the limits coming from parabolic regularization subject to the two different kinds of transmission conditions are different.
To show this consider the simple case of a junction with one incoming and one outgoing edges. So we have the conservation law on the incoming edge and on the outgoing one. Assume that there exists 0 <ρ <ρ < 1 and G > 0 such that f 1 (ρ) = f 2 (ρ) = G(ρ −ρ).
Consider the simplified version of (5) The unique solution of (13) is Therefore, as ε → 0 we get the solution of (10)-(11) This stationary solution is not admissible in the sense of the classical vanishing viscosity germ, see [5,Sec. 5], as it consists of a nonclassical shock. However, when dealing with conservation laws with discontinuous flux, it is well known that infinitely many L 1 contractive semigroups of solutions exist, also in relation with different physical applications. In particular, when the right and left fluxes are bell-shaped, as we assume in condition (H.1), each of those notions of admissible solution is uniquely determined by the choice of a (A, B)-connection, see [1,5,9,12] for precise definitions and exemples. In the exemple above the couple (ρ,ρ) is a connection.
It is worth noticing that entropy solutions admissible in the sense of a (A, B)connection can be obtained as limits of a sequence of parabolic approximations made with adapted viscosities but a classical condition of continuity at the interface, see [5, Sec. 6.2] for a general result, but also [2,15] for an application to the Buckley-Leverett equation.
It is difficult, however, to establish a direct equivalence between the aforementioned results and the one we put forward in this paper. In particular, in the present case we miss information on the boundary layers at the parabolic level and we do not know how the transmission conditions we impose on the parabolic fluxes translates into a condition for the hyperbolic problem.
This means in particular that we have little information on the germ associated to the family of limit solutions obtained in Theorem 1.2 and, so far, we have not been able to prove that this germ is L 1 -dissipative. We conjecture, however, that this is due to a technical obstruction and that uniqueness of the limit solutions holds.
The paper is organized as follows: Section 2 contains the precise list of assumptions on the initial and transmission conditions in the parabolic problem (5). In Section 3 we present the proofs of all necessary a priori estimates on (5). Finally, in Section 4 we detail the proof of Theorem 1.2.
Once the functions ρ ,0 are fixed, we impose on (5) initial conditions ρ ,0,ε such that for some constant C > 0 independent on ε.
The functions β appearing in the transmission conditions in (5) take the form for i ∈ {1, . . . , m}, and for j ∈ {m + 1, . . . , m + n} The In particular, (19) implies where χ (−∞,0) is the characteristic function of the set (−∞, 0). This specific form of transmission conditions is reminiscent of the parabolic transmission conditions considered in [14,8], which were originally inspired from the Kedem-Katchalsky conditions for membrane permeability introduced in [16] for some constants c h, > 0. Our conditions are more general and in particular we can notice that the function G h, above satisfies that allows the authors in [14] to get the L 2 conservation (see Lemma 3.3 below). We can observe that the equality (6) holds as and analogously m+n j=m+1 3. A priori estimates. This section is devoted to establish a priori estimates, uniform with respect to ε, which are necessary toward the proof of our main convergence result in the next section. For every ε > 0, let (ρ 1,ε , ..., ρ m+n,ε ) be a solution of (5) satisfying (16).
for every t ≥ 0.
for every t ≥ 0.