So welcome everyone. We have today Professor Serge Niquet from the Université Polytechnique

de France. And the title of the talk is quite long, maybe you can read it directly from

the slides. So Professor Niquet, please, we're looking forward to your talk.

Thank you. So first of all, I would like to thank you for the nice invitation. It's the

first time that I will present an online seminar, so I hope it will be okay. So as you see, it's a

joint work with Claire Shade from the University of Nice. This is in the south of France. And as you

see, I will speak about some model related to nanophotonics, and I will show you some analytical

results and also some numerical results. So, oh, oh it goes now. I have to do like that. Okay, so

the outline of my talk is the following one. After a short motivation, I will present you the model,

then I will consider or show that this model, this problem is well posed, then I will prove some

polynomial energy decay, and finally I will show you a numerical scheme and some numerical results.

So the motivation is actually nanophotonics, and more precisely I would like to consider

to control the interaction of lights, or this is physicians that wants to do so, they want to

control the interaction of light with nanoscale structures. And if the structure is metered,

then this is named nanoplasmonics. And actually the situation is that you have some electrons in a

middle, and then the electric fields act on this on this middle, and then we get a collective

oscillation of the electrons that then they form some clouds, and even some plasmons could appear.

So, and I have to do like this. So the main challenges is to prove some stability issues,

because this is an important feature in regards to a complete understanding of the phenomenon,

and actually it has also some impact on the development of adaptive numerical frameworks.

No, okay. So the model is the following one. We have some electrons in the meters, so we consider

it as a gas. So we use what they called in physics the hydrodynamical descriptions. This means that

if you assume that that n is the density of electrons and v the speed of these electrons,

then we have the non-linear transport equations in v and n, which is here, yeah.

And you see here that the E and H are involved, where E is the electric field, or and H is the

magnetic field. Mu is a constant, is the permeability, yes, I think, and M and gamma are constant.

And actually, these transport equations are coupled with the time domain Maxwell equations

via a source G, and G is given by this expression. This is minus E, and this is the

source G, and G is given by this expression. This is minus E and v. And let us notice that actually

this term is a quantum pressure term. And now this is fully non-linear. To get a linear model,

we use an argument that Claire used with our co-authors in that paper. We use a formal

linearization around an equilibrium state, namely, the equilibrium states will be n0.

For v0, we take 0, and we take, of course, an equilibrium for the electric and magnetic fields.

Well, n0, this is not written there, but it is supposed to be positive. And then,

if once we have these equilibrium states, we hold the new unknown small e, not boldface,

which is the difference between the equilibrium states and the states. And the same for H,

for n, we call it Q here. And the new unknowns, we replace v by G, which is minus n0 eV.

And if we fix some constant appropriately, we get the following system.

If omega represents the piece of middle, if we consider all the space time domain omega times

0 infinity, we get first the Maxwell equations here in E and H with a source term minus G here.

And this is coupled with these two equations in G and Q. And we may notice that if beta is different

from 0 here, we have the variable G is called the polarization current. And we see that we have a

PDE here. On the contrary, if beta equal to 0, then actually Q disappeared from these equations.

So we can decouple the, well, in a sense, the third, the fourth line or the fourth equation is

fully decoupled with the third first ones. And then we have here a model that is called,

that is named, then we have an ODE in G. And then this model is called the cold plasma model.

And I studied this model some years ago. And I proved a polynomial decay

in that paper. So of course, today, I will present you the case where beta is different from 0. So

the case where here we have a PDE. Note also that all constant here, beta is different from 0. So

beta squared is positive and all constant there are supposed to be positive.