Level set volume preserving diffusions
Yann Brenier (École Polytechnique)
MSRI: Simons Auditorium
We discuss diffusion equations that are constrained to preserve the volume
of each level set during the time evolution (which excludes the standard
heat equation). We consider, in particular, the gradient flow of the
Dirichlet integral under suitable volume-preserving transportation metrics.
The resulting equations are non-linear and very degenerate, admitting as
stationary solutions all scalar functions which are functions of their own
Laplacian. (In particular, in 2D, all stationary solutions of the Euler
equations for incompressible fluids.) We relate them to both combinatorial
optimization and linear algebra, through the quadratic assignmemt problem (a
NP combinatorial optimization problem including the travelling salesman
problem) and the Brockett-Wegner diagonalizing flow for linear operators.
For these equations, we provide a concept of "dissipative solutions" that
exist globally in time and are unique as long as they stay smooth, following
some works of P.-L. Lions (for the Euler equations) and
Ambrosio-Gigli-Savare (for the heat equation in metric spaces).