Arnold stability

Notes on the large time of Euler equations and inviscid damping

ttn12 fluid dynamics , ,

Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity {\omega = \omega(t,x,y)},

\displaystyle \partial_t \omega + (u \cdot \nabla) \omega =0

posed on a spatial domain {\Omega \subset \mathbb{R}^2}, where the velocity field {u\in \mathbb{R}^2} is obtained through the Biot-Savart law {u = \nabla^\perp \Delta^{-1} \omega}  (with convention (a_1,a_2)^\perp = (a_2,-a_1)). By construction, the velocity field {u} is incompressible: {\nabla \cdot u=0}. When dealing with domains with a boundary, {\Delta^{-1} } is defined together with a Dirichlet boundary condition that corresponds to the no-penetration condition of fluids {u\cdot n =0} 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 {L^p} norms of vorticity are conserved), and (2) weak solutions with bounded vorticity are unique (see Yudovich ’63).

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