Global, Local and Dense Non-mixing of the 3D Euler Equation
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Global, Local and Dense Non-mixing of the 3D Euler Equation Boris Khesin, Sergei Kuksin & Daniel Peralta-Salas Communicated by V. Šverák
Abstract We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a “typical” steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions u 0 of the Euler equation on S3 and divergence-free vector fields v0 arbitrarily close to u 0 , whose (non-steady) evolution by the Euler flow cannot converge in the C k Hölder norm (k > 10 non-integer) to any stationary state in a small (but fixed a priori) C k -neighbourhood of u 0 . The set of such initial conditions v0 is open and dense in the vicinity of u 0 . A similar (but weaker) statement also holds for the Euler flow on T3 . Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.
1. Introduction The dynamics of an ideal fluid on a Riemannian manifold is described by the Euler equation. It is an infinite-dimensional Hamiltonian system and has many peculiar properties, impossible in their finite-dimensional counterparts. For instance, in 2D the Euler equation possesses wandering solutions, which never return to the vicinity of the initial condition [20,22], while the Poincaré recurrence theorem would guarantee the return in finite-dimensional systems with convex Hamiltonians. In 3D the Euler equation has a global non-mixing property: there are two open sets of fluid velocity fields such that the solutions with initial conditions from one of these sets will never enter the other set [15]. In the present paper we prove that the non-mixing property has a local (and dense) nature: such two sets can be found in any neighbourhood of a “typical" steady solution, as we explain below. Thus
B. Khesin et al.
this does not only establish the ubiquitous appearance of non-mixing in the phase space, but this can also be thought of as a step towards the wandering property of Euler solutions in 3D, with the existence of solutions non-returning to a nearby neighbourhood of the initial conditions, rather than its own. Recall that the dynamics of an ideal fluid flow on a Riemannian 3-manifold M is described by the Euler equation on the fluid velocity field u(·, t): ∂t u + ∇u u = −∇ p , divu = 0 .
(1.1)
Here ∇u u is the covariant derivative of the vector field u along itself, div is the divergence operator computed with the Riemannian volume form, and p(·, t) is the pressure function, uniquely defined by the equations up to a constant. For a closed manifold M (i.e., compact and without boundary), the Euler equation defines a local flow {St } in the Hölder space of C k divergence-free vector fields on M provided that k > 1 is not an integer [7,10]. Accordin
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