Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity ,

posed on a spatial domain , where the velocity field is obtained through the Biot-Savart law . By construction, the velocity field is incompressible: . When dealing with domains with a boundary, is defined together with a Dirichlet boundary condition that corresponds to the no-penetration condition of fluids on the boundary.

The global well-posedness theory of 2D Euler is classical: (1) smooth initial data give rise to solutions that remain smooth for all times (e.g. the Beale-Kato-Majda criterium holds, as vorticity is uniformly bounded for all times; in fact, being transported along the volume-preserving flow, all norms of vorticity are conserved), and (2) weak solutions with bounded vorticity are unique (see Yudovich ’63).

The major open problem is to understand the large time behavior of solutions to the 2D Euler equations. This is notoriously difficult, being time-reversible Hamiltonian system and having conserved energy and many invariant Casimir’s

that are conserved for all times, for any reasonable function . A great reference that discusses in depth this topics and other questions in fluid dynamics is this lecture notes by V. Sverak.

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