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| Magali Mercier Mail: mercier(at)math.univ-lyon1(.)fr |
I am working on fluid mechanics. More precisely, I am searching shock waves solutions for the Euler
compressible equation when the discontinuity is a compact manifold.
For example, we are interested
in the case of spherical symmetry. We already know that we have local in time
existence; however, we do not know so much thing about the lifespan of the solution in the general case.
We would like to obtain such solutions for long times, for example by gluing classical solutions
along a line of discontinuity whom position is given thanks to the Rankine-Hugoniot conditions.
In the spherically symmetric case it seems not so difficult, but in fact, we have trouble when we search
classical solutions with a large time of existence.
An other approach consist in stting an equation on the position of the discontinuity. We obtain
a 'mixt' problem. We would like, in the hyperbolic domain, to make a change of variable in order to get a
non-linear wave equation.
For the scalar conservation laws, we have at disposal the L1 framework.
In particular, this theory is based on the Kruzkov theorem, what says in particular how the solutions
depends on the initial conditions. This question is important in numerical analysis, for example.
With Rinaldo M. Colombo, we have generalised this theorem by studying the dependence to flow and source terms.
This new result allowed us then to obtain global existence of solution for the 'toy'-model
of radiating gas.
We have also studied the control of the continuity equation with a non-local flow and the interaction between a group and an individual.
Here, I have studied a model of roundabout and solved the Riemann problem for this model. The next step could be the Cauchy problem, but the Riemann soler is not continuous when the densities are maximal, what make us wonder if the model is really good.