Okay, so now we are continuing with Nora Philippi, also from the Technical University of Darmstadt,

who is also working on her PhD thesis now.

And she will be speaking on the transport limit of singularity, better convection, the efficient

problems on networks.

Please, Nora.

So, hello everyone.

Thank you, Marius, for this kind introduction.

So yeah, my talk is about the transport limit of singularity, better convection, diffusion

problems on networks, which is a joint work with my supervisor, Herbert Egger.

And first of all, I want to thank the organizers for giving me the opportunity to speak here

at this mini workshop.

And maybe also as a little background information, I'm just as Elisa, also part of the Collaboration

Research Center, which is dealing with modeling, simulation and optimization using the example

of gas networks.

So also here, I will be dealing with partial differential equations on networks.

So let's get started.

So first of all, I will introduce the convection diffusion problem as well as its limiting

transport problem on a network of pipes.

And this network is described by a one-dimensional metric graph.

And additionally to the partial differential equations on each pipe of the network, we

need some coupling conditions at pipe junctions.

And I will also introduce some basic properties and we will discuss the well-posedness of

those problems.

And after that, we come to the main part of my talk, which is about the asymptotic analysis.

We are interested in the vanishing diffusion case.

And first of all, we will consider this problem on a single pipe, which has been intensively

studied in the literature, so it's nothing new.

And we will have a look at boundary layers that will occur and at some asymptotic convergence

results.

And the main contribution that we did here is the extension of this results to network

and we will also have a brief look into that.

But first of all, we will start with the convection diffusion problem.

So as I said, we have a problem on a network, which is described here by a one-dimensional

graph.

And on each pipe, we consider the following convection diffusion equation, which is given

here, that is complemented by some initial conditions.

We assume that the parameters are positive and constant on each pipe and we also assume

that the parameter b sums up to zero at the pipe junction v3 here.

And the assumptions on these parameters can easily be relaxed.

We just need to assume that a and the phylum are uniformly positive and bounded by the

low and the parameter b should not vanish.

And this problem, for instance, describes the transport and diffusion of some salt contaminant

in the flow of water through a network of pipes, where in this case, the parameter a

is the cross-section of each pipe, the parameter b is the volume flow rate of the water in

the network, and the parameter at phylum is the diffusion coefficient.

But additionally to the partial differential equation on each pipe, we need some boundary

conditions at the boundary vertices and some coupling conditions at the interior vertex

v3 here in our example.

So we assume, or we know that we need one boundary condition at each end of the pipe

for this convection diffusion problem.