So thank you very much for the nice invitation to this seminar. It's really a pleasure for

me to contribute to this set of presentations. And I will talk today about conservation law

on a star-shaped network. So let's get started with a description of the geometry that interests

us today. So we consider a very simple network consisting of only one central node and then

a finite number of edges, which are half-lines. So we parameterize them so that some of them

are called incoming edges. They are parameterized by x belonging to R minus, and some of them

are outgoing edges. So they are parameterized by the positive half of the real axis. And

in this way, we have that the central node is always at x equal to zero for each of the

nodes. So the dynamic on this network is described by hyperbolic conservation laws, scalar. We

have a prototype of them here in this formula. So we have essentially that rho is the density

of any extensive physical variable that you can imagine. And we assume that it is conserved.

So there is no production or disappearance of rho during the evolution. So conservation

laws, scalar conservation laws, have a very well-established theory in, well, when the

domain is R, and they have two very particular properties that are very useful in our talk.

So first of all, all of the waves in the solution propagates with finite speed. And this means

that even if we consider a very special kind of network, the star-shaped network that we

have in the first slide, this can be considered as a building block for more complicated networks.

Because due to the finite speed of propagation of wavefront, we have that two separate nodes

do not interact immediately. Actually, their dynamic for a smaller interval of time is

rather independent. So the second property that is very important in our discussion is

the fact that even if we start with extremely smooth initial condition, we can have formation

of discontinuity in the solution. So for this reason, for us, when I talk about solution,

I'm not talking about classical solutions. So they are not functions which are differentiable

in respect to their variable, but they are weak solutions. And it happens that weak solutions,

so solution in the sense of distribution, would be too many. It is not possible to have

a well-posed theory for them because we lose uniqueness, essentially. So we have to introduce

a further condition to have a family of solution for which we can establish well-posedness.

And these conditions are called entropy conditions. So we will go back to that later on, but it

is really important to know that weak solutions are not unique for conservation laws, and

we need to add extra conditions to get well-posedness.

So the other condition that I'm putting here on the flux are basically just to simplify

the situation in the torque. Essentially, the shape of flux we consider is the one which

is here in the figure. We need to have just one maximum somewhere, and the flux is zero

at zero and at point capital R, or well, one in this drawing. So since we are considering

conservation laws, so we have this idea of conserving quantity, we have that at any time,

the total amount of rho on the network should be the same. So we should have this equality

satisfied. And if we, so essentially, this condition translates into a balance for the

trace of the fluxes at the node. So very often in my talk, I will refer to the conservation

condition at the junction by thinking to this equality here. So the sum of incoming fluxes

should be the same as the sum of outgoing fluxes.

So I was telling you that we have a concept of weak solution and the concept of entropy

solution for a conservation law defined on the real axis. So on a network, we have something

similar in a way. So we consider a weak solution on the network as twice entropy admissible.

So it is a vector of function, each of them is defined on one of the edges. It is a Krutschk-entropy

solution in the interior of its own edges. So it satisfies this kind of inequality for

any k here, you see the constant k, and for any test function which is positive. And of

course, with support is strictly contained inside omega h. And moreover, we want this

conservation at the junction. So we have that at the level of the junction, we are just

enforcing conservation, which is a property which relates to weak solution, but does not

take into consideration some more selective criteria like an entropy condition. And because