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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 $mathbb S^3$ and divergence-free vector fields $v_0$ arbitrarily close to $u_0$, whose (non-steady) evolution by the Euler flow cannot converge in the $C^k$ Holder 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 $v_0$ is open and dense in the vicinity of $u_0$. A similar (but weaker) statement also holds for the Euler flow on $mathbb T^3$. 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.
The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [35,36] launched a programme to address the global existence problem for the Euler and Navier Stokes equations based
The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $mathbb{R}^n to mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic features of the
The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation begin{equation}label{eq:discount 1}tag{HJ$_lambda$} lambda v(x)+H( x, Dv(x) )=0 , quad xin mathbb{R}^n.
Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this re
We review the continuous symmetry approach and apply it to find the solution, via the construction of constants of motion and infinitesimal symmetries, of the 3D Euler fluid equations in several instances of interest, without recourse to Noethers the