So welcome everyone to this CAA online seminar. Today we have Professor Christian Hickenberg

from the University of Pittsburgh here in Germany. And he will be speaking about finding

solutions of the multi-dimensional compressive equations. Please, you have the floor. Thank

you. Yes, thank you for having me. Thank you for the introduction. Thank you for having

me in Erlangen. It's a great privilege. So I'm from Würzburg, which is the neighboring

university. And this is joint work with Vasily Barsukov. So he's a postdoc in Zurich. He

got his PhD with me. And then the lecture has two parts. There's somewhat a little separate.

And the second part is with Simon Markfeld and others who just started his postdoc at

Cambridge University. And so Phil Rowe was involved with Vasily and myself, and Edward

Feireiser was involved with Simon Markfeld and myself. So here we go. So in 1757, Euler

was the first person to write down the Euler equations, what are called the Euler equations

today. I write them here in today's notation, conservation of mass, conservation of total

momentum. And you notice he wrote them down in multi-D and 3D. He wrote down the compressible

versions of the version of the equations. So then about 100 years later, Riemann attempted

to solve them while he found a special case where he could actually solve them. It physically

corresponds to a tube with a membrane in it, two constant states on the left and the right,

the so-called Riemann problem. And it gives rise to jumps that are called shock waves,

other types of jumps called contact discontinuities, and these continuous transitions called rarefaction

waves. And so here you see it in the X-T plane. It's a self-similar solution. So here we have

an initial data, one jump into another jump, giving rise to shock, contacts, and rarefaction

in such a fashion. So this is what he could do. And then in the middle of the 20th century,

aeronautics aerospace demanded good solutions of the Euler equations. And so in Russia,

Godunov, I think he used this scheme in the early 50s, but he published it in 59, a numerical

scheme for the Euler equations called the Godunov scheme. So here you have the X-axis.

He has, suppose you have initially some density, it's called Q here, which first you approximate

by piecewise constant. Now here you have the X and the T-axis. So you evolve this. And

so he uses this Riemann from a hundred years earlier. So he gets an exact evolution of

this approximate data, initial data, which gives rise to a solution, say something like

that at some later time. This is X. Now we're back at the solution. In order to iterate

this, this solution then gets projected back down to piecewise constant. And so he can

do this again. So in 1D, this is a wonderful scheme. It incorporates the non-linearity

via these Riemann problems. And so Peter Lacks caught on to this. So now we're going from

Novosibirsk to New York. And he asked if one could obtain the existence result for the

1D Euler equations by proving convergence of Godunov's method. He asked it slightly

more general, but this is not a bad way to put it because he didn't see how to do it.

And actually a young postdoc who was there at Quran, Jim Glimm in 1964 then published

or 65 published a paper where he proved convergence, but he had to change the Godunov method a

little bit. But in principle, it's something like that. He had to introduce a probabilistic

element. And here's Glimm himself. And so he was able to prove for systems of conservation

laws of one space dimension. The Euler equations are an example of that. And so you can see

these Riemann solutions. You can see the time going up. And it was a fabulous piece of work.

So he found this sequence of approximate solutions named the approximation being this Riemann

problem. I'm sorry, this numerical scheme. Here it's called the Glimm scheme. It's the

variant of Godunov's. Then he proved boundedness in the right norm and showed convergence actually

to a weak solution of the conservation laws. I mean, these also for other, as I showed

you, the solutions in general will not be continuous. So you need, they may have jumps.

So you introduce this notion of weak solution. And in that he'd managed to prove convergence.

So it's an amazing piece of work. And the subject, the theoretical aspect of the subject

has been deeply influenced by that since then. But it's in one space dimension. And so another

assumption that he needed is he needed that the initial data had small bounded variation.