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We study so-called supercritical mean-field limits of systems of trapped particles moving according to Newton's second law with either Coulomb/super-Coulomb or regular interactions, from which we derive a $\mathsf{d}$-dimensional generalization of the Lake equation, which coincides with the incompressible Euler equation in the simplest setting, for monokinetic data. This supercritical mean-field limit may also be interpreted as a combined mean-field and quasineutral limit, and our assumptions on the rates of these respective limits are shown to be optimal. Our work provides a mathematical basis for the universality of the Lake equation in this scaling limit -- a new observation -- in the sense that the dependence on the interaction and confinement is only through the limiting spatial density of the particles. Our proof is based on a modulated-energy method and takes advantage of regularity theory for the obstacle problem for the fractional Laplacian., Comment: 41 pages

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows., Comment: 69 pages

This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Gr\"onwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper., Comment: 30 pages

We consider overdamped Langevin dynamics for the attractive log gas on the torus $\mathbb{T}^\mathsf{d}$, for $\mathsf{d}\geq 1$. In dimension $\mathsf{d}=2$, this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature $\beta_{\mathrm{c}}={2\mathsf{d}}$ corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold $\beta_{\mathrm{s}}$ sharply corresponding to the nonlinear stability of the uniform distribution. We show that for $\beta>\beta_{\mathrm{s}}$, the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for $\beta<\beta_{\mathrm{s}}$, it is stable. We also show that there exists $\beta_{\mathrm{u}}$ for which uniqueness of equilibria holds for $\beta<\beta_{\mathrm{u}}$. We establish a uniform-in-time rate for entropic propagation of chaos for a range of $\beta<\beta_{\mathrm{s}}$. To our knowledge, this is the first such result for singular attractive interactions and affirmatively answers a question of Bresch et al. arXiv:2011.08022. The proof of the convergence is through the modulated free energy method, relying on a modulated logarithmic Hardy-Littlewood-Sobolev (mLHLS) inequality. Unlike Bresch et al., we show that such an inequality holds without truncation of the potential -- the avoidance of the truncation being essential to a uniform-in-time result -- for sufficiently small $\beta$ and provide a counterexample to the mLHLS inequality when $\beta>\beta_{\mathrm{s}}$. As a byproduct, we show that it is impossible to have a uniform-in-time rate of propagation of chaos if $\beta>\beta_{\mathrm{s}}$., Comment: 69 pages

We consider mean-field limits for overdamped Langevin dynamics of $N$ particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential-in-time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-$N$ "modulated logarithmic Sobolev inequality" holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension., Comment: 19 pages

Motivated by the possibility of noise to cure equations of finite-time blowup, recent work arXiv:2109.09892 by the second and third named authors showed that with quantifiable high probability, random diffusion restores 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 the Patlak-Keller-Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from arXiv:2109.09892 in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow\infty$., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2109.09892

We consider conservative and gradient flows for $N$-particle Riesz energies with mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range $d-2\leq s

Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension $d\ge 2$, the smallness of the nonlinearity $\lambda$ is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. In the present work, we build on the framework of [ST21], following the formal derivation of Spohn [Spo06] and inspired by previous work [HO21] of the first author and Olla, so that the inhomogeneous 3-wave kinetic equation can also be obtained at the kinetic limit under analogous assumptions. Similar to the homogeneous case -- and unlike the cubic nonlinear Schr\"odinger equation -- the inhomogeneous kinetic description of the deterministic lattice ZK equation is unlikely to happen due to the vanishing of the dispersion relation on a certain singular manifold on which not only $3$-wave interactions but also all $n$-wave interactions ($n\ge3$) are allowed to happen, a phenomenon first observed by Lukkarinen [Luk07]. To the best of our knowledge, our work provides the first rigorous derivation of a nonlinear inhomogeneous wave kinetic equation in the kinetic limit., Comment: arXiv admin note: substantial text overlap with arXiv:2106.09819

We consider the Vlasov equation in any spatial dimension, which has long been known to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie-Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for 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 arXiv:1908.03847, which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schr\"odinger 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 on providing a "statistical basis" for the bracket structure of the Vlasov equation., Comment: 54 pages

We give a rigorous, quantitative derivation of the incompressible Euler equation from the many-body problem for $N$ bosons on $\mathbb{T}^d$ with binary Coulomb interactions in the semiclassical regime. The coupling constant of the repulsive interaction potential is $~1/(\varepsilon^2 N)$, where $\varepsilon \ll 1$ and $N\gg 1$, so that by choosing $\varepsilon=N^{-\lambda}$, for appropriate $\lambda>0$, the scaling is supercritical with respect to the usual mean-field regime. For approximately monokinetic initial states with nearly uniform density, we show that the density of the first marginal converges to 1 as $N\rightarrow\infty$ and $\hbar\rightarrow 0$, while the current of the first marginal converges to a solution $u$ of the incompressible Euler equation on an interval for which the equation admits a classical solution. In dimension 2, the dependence of $\varepsilon$ on $N$ is essentially optimal, while in dimension 3, heuristic considerations suggest our scaling is optimal. Our proof is based on a Gronwall relation for a quantum modulated energy with an appropriate corrector and is inspired by recent work of Golse and Paul arXiv:1912.06750 on the derivation of the pressureless Euler-Poisson equation in the classical and mean-field limits and of Han-Kwan and Iacobelli arXiv:2006.14924 and the author arXiv:2104.11723 on the derivation of the incompressible Euler equation from Newton's second law in the supercritical mean-field limit. As a byproduct of our analysis, we also derive the incompressible Euler equation from the Schr\"odinger-Poisson equation in the limit as $\hbar+\varepsilon\rightarrow 0$, corresponding to a combined classical and quasineutral limit., Comment: 29 pages

Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. 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. arXiv:1806.03734 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 Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability., Comment: 34 pages

We consider the mean-field limit of systems of particles with singular interactions of the type $-\log|x|$ or $|x|^{-s}$, with $0< s0$, the convergence is global in time, and it is the first such result valid for both conservative and gradient flows in a singular setting on $\mathbb{R}^d$. The proof relies on an adaptation of an argument of Carlen-Loss to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in arXiv:1508.03377, arXiv:1803.08345, arXiv:2107.02592 making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution., Comment: 41 pages

We provide a proof of mean-field convergence of first-order dissipative or conservative dynamics of particles with Riesz-type singular interaction (the model interaction is an inverse power $s$ of the distance for any $0

A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli arXiv:2006.14924 showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of $N$ particles interacting in $\mathbb{T}^d$, $d\geq 2$, via Newton's second law through a supercritical mean-field limit. Namely, the coupling constant $\lambda$ in front of the pair potential, which is Coulombic, scales like $N^{-\theta}$ for some $\theta \in (0,1)$, in contrast to the usual mean-field scaling $\lambda\sim N^{-1}$. Assuming $\theta\in (1-\frac{2}{d(d+1)},1)$, they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as $N\rightarrow\infty$. Han-Kwan and Iacobelli asked if their range for $\theta$ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit $N\rightarrow\infty$ for $\theta \in (1-\frac{2}{d},1)$. For reasons of scaling, this range appears optimal in all dimensions. Our proof is based on Serfaty's modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved "renormalized commutator" estimate to obtain the larger range for $\theta$., Comment: 20 pages

In arXiv:1201.4067 and arXiv:1611.08030, Eyink and Shi and Chibbaro et al., respectively, formally derived an infinite, coupled hierarchy of equations for the spectral correlation functions of a system of weakly interacting nonlinear dispersive waves with random phases in the standard kinetic limit. Analogously to the relationship between the Boltzmann hierarchy and Boltzmann equation, this spectral hierarchy admits a special class of factorized solutions, where each factor is a solution to the wave kinetic equation (WKE). A question left open by these works and highly relevant for the mathematical derivation of the WKE is whether solutions of the spectral hierarchy are unique, in particular whether factorized initial data necessarily lead to factorized solutions. In this article, we affirmatively answer this question in the case of 4-wave interactions by showing, for the first time, that this spectral hierarchy is well-posed in an appropriate function space. Our proof draws on work of Chen and Pavlovi\'{c} for the Gross-Pitaevskii hierarchy in quantum many-body theory and of Germain et al. for the well-posedness of the WKE., Comment: 25 pages

In arXiv:1004.1407, Flandoli, Gubinelli, and Priola proposed a stochastic variant of the classical point vortex system of Helmholtz and Kirchoff in which multiplicative noise of transport-type is added to the dynamics. An open problem in the years since is to show that in the mean-field scaling regime, in which the circulations are inversely proportional to the number of vortices, the empirical measure of the system converges to a solution of a two-dimensional Euler vorticity equation with multiplicative noise. By developing a stochastic extension of the modulated-energy method of Serfaty and Duerinckx for mean-field limits of deterministic particle systems and by building on ideas introduced by the author for studying such limits at the scaling-critical regularity of the mean-field equation, we solve this problem under minimal assumptions., Comment: 34 pages

We consider first-order conservative systems of particles with binary Coulomb interactions in the mean-field scaling regime in dimensions $d\geq 3$. We show that if at some time, the associated sequence of empirical measures converges in a suitable sense to a probability measure with bounded density $\omega^0$ as the number of particles $N\rightarrow\infty$, then the sequence converges for short times in the weak-* topology for measures to the unique solution of the mean-field PDE with initial datum $\omega^0$. This result extends our previous work arXiv:2004.04140 for point vortices (i.e. $d=2$). In contrast to the previous work arXiv:1803.08345, our theorem only requires the limiting measure belong to a scaling-critical function space for the well-posedness of the mean-field PDE, in particular requiring no regularity. Our proof is based on a combination of the modulated-energy method of Serfaty and a novel mollification argument first introduced by the author in arXiv:2004.04140., Comment: 32 pages. arXiv admin note: text overlap with arXiv:2004.04140

We consider the classical point vortex model in the mean-field scaling regime, in which the velocity field experienced by a single point vortex is proportional to the average of the velocity fields generated by the remaining point vortices. We show that if at some time, the associated sequence of empirical measures converges in a suitable sense to a probability measure with density $\omega^0\in L^\infty(\mathbb{R}^2)$ and having finite energy, as the number of point vortices $N\rightarrow\infty$, then the sequence converges in the weak-* topology for measures to the unique solution $\omega$ of the 2D incompressible Euler equation with initial datum $\omega^0$, locally uniformly in time. In contrast to previous results, our theorem requires no regularity assumptions on the limiting vorticity $\omega$, is at the level of conservation laws for the 2D Euler equation, and provides a quantitative rate of convergence. Our proof is based on a combination of the modulated-energy method of Serfaty and a novel mollification argument. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich for global well-posedness of 2D Euler in $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$., Comment: 47 pages; updated discussion on prior results and added an acknowledgement

We consider the well-known Lieb-Liniger (LL) model for $N$ bosons interacting pairwise on the line via the $\delta$-potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schr\"odinger equation (NLS) with an explict rate of convergence. In contrast to previous work arXiv:0906.3047 relying on quantum field theory and without an explicit rate, our proof is inspired by the counting method of Pickl arXiv:0907.4464 and Knowles and Pickl arXiv:0907.4313. To overcome difficulties stemming from the singularity of the $\delta$-potential, we introduce a new short-range approximation argument that exploits the H\"older continuity of the $N$-body wave function in a single particle variable. By further exploiting the $L^2$-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence assuming only that the limiting solution to the NLS has finite mass., Comment: 32 pages; strengthened result to allow for finite mass solutions; deleted background material; added references; updated acknowledgements; corrected typos

We consider the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. We establish the existence of an infinite sequence of observables such that the corresponding trace functionals, which we call ``energies,'' commute with respect to the weak Lie-Poisson structure defined by the authors in arXiv:1908.03847. The Hamiltonian equation associated to the third energy functional is precisely the GP hierarchy. The equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schr\"odinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schr\"odinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system., Comment: 97 pages

We consider the cubic nonlinear Schr\"odinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schr\"odinger equation from quantum many-body systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein for a system describing the evolution of finitely many indistinguishable classical particles., Comment: 81 pages

In this paper, we give a rigorous justification of the point vortex approximation to the family of modified surface quasi-geostrophic (mSQG) equations globally in time in both the inviscid and vanishing dissipative cases. This result completes the justification for the remaining range of the mSQG family unaddressed by Geldhauser and Romito in the article arXiv:1812.05166, in the case of identically signed vortices., Comment: 35 pages

In this paper, we prove global well-posedness and scattering of the Cauchy problem for the elliptic-elliptic Davey-Stewartson system (eeDS) for initial data $u_{0}\in L^{2}(\mathbb{R}^{2})$ in the defocusing case and for $u_{0}\in L^{2}(\mathbb{R}^{2})$ with mass below that of the ground state in the focusing case. This result resolves the large data problem at the scaling-critical regularity left open by Ghidaglia and Saut in their work initiating the mathematical study of the Cauchy problem for the system. Our proof uses the concentration compactness/rigidity road map of Kenig and Merle together with the long-time Strichartz estimate approach of Dodson. Due to the failure of the endpoint $L_{t}^{2}L_{x}^{\infty}$ Strichartz estimate, we rely heavily on bilinear Strichartz estimates. We overcome the obstruction to applying such estimates caused by the lack of permutation invariance of the eeDS nonlinearity under frequency decomposition by introducing a new frequency cube decomposition of the nonlinearity and proving bilinear estimates suited to this decomposition. In both the defocusing and focusing cases, we overcome the lack of an a priori interaction Morawetz estimate by exploiting the spatial and frequency localization of the minimal counterexamples which we reduce to considering., Comment: 129 pages