arXiv subjects

Matthew Rosenzweig

Publications and source records attributed to Matthew Rosenzweig.

35 records · Page 2Linked to original sources

arXiv2023

Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

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 $d-2\leq s<d$ not addressed by previous work of the second two authors. We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang for so-called $\dot{W}^{-1,\infty}$ kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al.

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arXiv2022

On the wave turbulence theory for a stochastic KdV type equation -- Generalization for the inhomogeneous kinetic limit

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 $λ$ 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 Schrödinger 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.

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arXiv2022

Mean-field limits of Riesz-type singular flows

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 $0<s<d$) when assuming a certain regularity of the solutions to the limiting evolution equations. It relies on a modulated-energy approach, as introduced in previous works where it was restricted to the Coulomb and super-Coulombic cases. The method is also capable of incorporating multiplicative noise of transport type into the dynamics. It relies in extending functional inequalities of arXiv:1803.08345, arXiv:2011.12180, arXiv:2003.11704 to more general interactions, via a new, robust proof that exploits a certain commutator structure.

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arXiv2022

A rigorous derivation of the Hamiltonian structure for the Vlasov equation

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 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 on providing a "statistical basis" for the bracket structure of the Vlasov equation.

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arXiv2021

From Quantum Many-Body Systems to Ideal Fluids

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 $\varepsilon=N^{-λ}$, for appropriate $λ>0$, 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 Schrödinger-Poisson equation in the limit as $\hbar+\varepsilon\rightarrow 0$, corresponding to a combined classical and quasineutral limit.

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arXiv2021

Global solutions of aggregation equations and other flows with random diffusion

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.

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arXiv2021

Global-in-time mean-field convergence for singular Riesz-type diffusive flows

We consider the mean-field limit of systems of particles with singular interactions of the type $-\log|x|$ or $|x|^{-s}$, with $0< s 0$, 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.

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arXiv2021

On the Rigorous Derivation of the Incompressible Euler Equation from Newton's Second Law

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 $λ$ in front of the pair potential, which is Coulombic, scales like $N^{-θ}$ for some $θ\in (0,1)$, in contrast to the usual mean-field scaling $λ\sim N^{-1}$. Assuming $θ\in (1-\frac{2}{d(d+1)},1)$, 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 $θ$ 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 $θ\in (1-\frac{2}{d},1)$. 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 $θ$.

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arXiv2021

Uniqueness of Solutions to the Spectral Hierarchy in Kinetic Wave Turbulence Theory

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 Pavlović for the Gross-Pitaevskii hierarchy in quantum many-body theory and of Germain et al. for the well-posedness of the WKE.

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arXiv2020

The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise

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.

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arXiv2020

The Mean-Field Limit of the Lieb-Liniger Model

We consider the well-known Lieb-Liniger (LL) model for $N$ bosons interacting pairwise on the line via the $δ$-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 Schrödinger 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 $δ$-potential, we introduce a new short-range approximation argument that exploits the Hölder 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.

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arXiv2020

Mean-Field Convergence of Point Vortices without Regularity

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 $ω^0\in L^\infty(\mathbb{R}^2)$ 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 $ω$ of the 2D incompressible Euler equation with initial datum $ω^0$, locally uniformly in time. In contrast to previous results, our theorem requires no regularity assumptions on the limiting vorticity $ω$, 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 $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$.

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arXiv2020

Mean-Field Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity

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 $ω^0$ 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 $ω^0$. 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.

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arXiv2019

Poisson Commuting Energies for a System of Infinitely Many Bosons

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 Schrödinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schrödinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system.

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arXiv2019

A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schrödinger Equation

We consider the cubic nonlinear Schrödinger 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 Schrödinger 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.

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arXiv2019

Justification of the Point Vortex Approximation for Modified Surface Quasi-Geostrophic Equations

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.

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arXiv2018

Global Well-Posedness and Scattering for the Elliptic-Elliptic Davey-Stewartson System at $L^{2}$-Critical Regularity

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.

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