Seminar

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).