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Free, publiclyaccessible full text available April 1, 2024

Abstract A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, HanKwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of
N particles interacting in ,$${\mathbb {T}}^d$$ ${T}^{d}$ , via Newton’s second law through a$$d\ge 2$$ $d\ge 2$supercritical meanfield limit . Namely, the coupling constant in front of the pair potential, which is Coulombic, scales like$$\lambda $$ $\lambda $ for some$$N^{\theta }$$ ${N}^{\theta}$ , in contrast to the usual meanfield scaling$$\theta \in (0,1)$$ $\theta \in (0,1)$ . Assuming$$\lambda \sim N^{1}$$ $\lambda \sim {N}^{1}$ , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$\theta \in (1\frac{2}{d(d+1)},1)$$ $\theta \in (1\frac{2}{d(d+1)},1)$ . HanKwan and Iacobelli asked if their range for$$N\rightarrow \infty $$ $N\to \infty $ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$\theta $$ $\theta $ for$$N\rightarrow \infty $$ $N\to \infty $ . Our proof is based on Serfaty’s modulatedenergy method, but compared to that of HanKwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta \in (1\frac{2}{d},1)$$ $\theta \in (1\frac{2}{d},1)$ . Additionally, we show that for$$\theta $$ $\theta $ , one cannot, in general, expect convergence in the modulated energy notion of distance.$$\theta \le 1\frac{2}{d}$$ $\theta \le 1\frac{2}{d}$ 
Abstract Aggregation equations, such as the parabolicelliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finitetime blowup. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370–9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevreytype Fourier–Lebesgue spaces with quantifiable high probability.

We consider the wellknown LiebLiniger (LL) model for
bosons interacting pairwise on the line via the\begin{document}$ N $\end{document} potential in the meanfield scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the timedependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the onedimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [\begin{document}$ \delta $\end{document} 3 ] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65 ,66 ,67 ] and Knowles and Pickl [44 ]. To overcome difficulties stemming from the singularity of the potential, we introduce a new shortrange approximation argument that exploits the Hölder continuity of the\begin{document}$ \delta $\end{document} body wave function in a single particle variable. By further exploiting the\begin{document}$ N $\end{document} subcritical wellposedness theory for the 1D cubic NLS, we can prove meanfield convergence when the limiting solution to the NLS has finite mass, but only for a very special class of\begin{document}$ L^2 $\end{document} body initial states.\begin{document}$ N $\end{document}