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Abstract We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is ofLie–Poisson type. In parallel, it is classical that the Vlasov equation is amean-field limitfor a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to [MNP⁺20], which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison and Weinstein [MMW84] on providing a ‘statistical basis’ for the bracket structure of the Vlasov equation. 

We consider the 2D incompressible Euler equation on a bounded simply connected domain\Omega. We give sufficient conditions on the domain\Omegaso that for any initial vorticity\omega_{0} \in L^{\infty}(\Omega), the weak solutions are unique. Our sufficient condition is slightly more general than the condition that\Omegais aC^{1,\alpha}domain for some\alpha>0, with its boundary belonging toH^{3/2}(\mathbb{S}^{1}). As a corollary, we prove uniqueness forC^{1,\alpha}domains for\alpha >1/2and for convex domains which are alsoC^{1,\alpha}domains for some\alpha >0. Previously, uniqueness for general initial vorticity inL^{\infty}(\Omega)was only known forC^{1,1}domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below theC^{1,1}regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional. 

Abstract We consider the 2D incompressible Euler equation on a corner domain Ω with angle νπ with 1 2 < ν < 1 . We prove that if the initial vorticity ω 0 ∈ L 1 (Ω) ∩ L ∞ (Ω) and if ω 0 is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is non-constant around the boundary.