# Matthew Rosenzweig's Webpage

# About me:

My name is Matthew "Matt" Rosenzweig. I am a CLE Moore Instructor at MIT and postdoctoral associate of the Simons Collaboration on Wave Turbulence. I defended my dissertation at the University of Texas at Austin in April 2020, under the supervision of Professor Natasa Pavlovic. Previously, I completed my undergraduate degree at Harvard University. Here is a copy of my CV.

Email Address: mrosenzw@mit.edu

## Research Interests:

My current research interests are in nonlinear partial differential equations and mathematical physics. More specifically, I have been recently interested in the mathematics of nonlinear dispersive and fluid equations and their derivation from underlying physical problems, such as water waves and quantum many-body systems. More on my mathematical interests can be learned from my research statement.

## Papers and Preprints:

Full Justification of the Davey-Stewartson System from 3D Finite-Depth Gravity Water Waves, in preparation.

From Quantum Many-Body Systems to Ideal Fluids, in preparation.

A Rigorous Justification of the Stochastic Point Vortex Model with Multiplicative Noise, in preparation.

Global Solutions to Aggregation Equations and Other Flows with Random Diffusion, w/ G. Staffilani, in preparation.

Global-in-time Mean-Field Convergence for Singular Riesz-type Diffusive Flows, w/ S. Serfaty, https://arxiv.org/abs/2108.09878, 41 pgs., preprint (2021).

Mean-Field Limits of Riesz-Type Singular Flows with Possible Multiplicative Transport Noise, w/ Q.H. Nguyen and S. Serfaty, http://arxiv.org/abs/2107.02592, 40 pgs., preprint (2021).

On the Rigorous Derivation of the Incompressible Euler Equation from Newton's Second Law, https://arxiv.org/abs/2104.11723, 20 pgs., preprint (2021).

Uniqueness of Solutions to the Spectral Hierarchy in Kinetic Wave Turbulence Theory, https://arxiv.org/abs/2104.06907, 25 pgs., preprint (2021).

The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise, https://arxiv.org/abs/2011.12180, 34 pgs., preprint (2020).

Mean-Field Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity, https://arxiv.org/abs/2010.10009, 32 pgs., preprint (2020).

Mean-Field Convergence of Point Vortices without Regularity, https://arxiv.org/abs/2004.04140 , 47 pgs., preprint (2020).

Old and New Perspectives on Effective Equations: A Study of Quantum Many-Body Systems, dissertation, 398 pgs., (2020).

Global Well-Posedness and Scattering for the Hartree Equation at $L^{2}$-Critical Regularity, preprint, 137 pgs., (2019).

The Mean-Field Limit of the Lieb-Liniger Model, https://arxiv.org/abs/1912.07585, preprint, 32 pgs., (2019).

Poisson Commuting Energies for a System of Infinitely Many Bosons, w/ D. Mendelson A.R. Nahmod, N. Pavlovic, and G. Staffilani, https://arxiv.org/abs/1910.06959 , 97 pgs., preprint (2019).

A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schrodinger Equation, w/ D. Mendelson, A.R. Nahmod, N. Pavlovic, and G. Staffilani, https://arxiv.org/abs/1908.03847, 81 pgs., preprint (2019).

Rigorous Justification of the Point Vortex Approximation for Modified Surface Quasi-Geostrophic Equations, https://arxiv.org/abs/1905.07351, 35 pgs., preprint (2019).

Global Well-Posedness and Scattering for the Elliptic-Elliptic Davey-Stewartson System at $L^{2}$-Critical Regularity, https://arxiv.org/abs/1808.01955, 129 pgs., preprint (2018).

## Notes and Expository Material:

### Caveat Emptor: the material below is likely riddled with typos.

Beyond Gronwall for gradient flows - A short note showing that in some cases (e.g. Coulomb gradient flow on $\mathbb{R}^d$), one can prove mean-field convergence for noiseless systems of particles with a convergence rate having time factor $O(t^{\delta})$, for some $\delta>0$. While well-known that noise improves the growth in time of the rate (possibly making the rate uniform), it doesn't seem to have been previously observed you can go beyond the Gronwall $e^{Ct}$ growth without noise.

Pickl's method for $H^1$ data - A note showing how to use dispersion of the limiting equation to reduce the regularity assumptions in Pickl's derivation of the 3D Gross-Pitaevskii equation.

Convolution inequalities for Boltzmann collision operator - These notes are based on the work by Alonso, Carneiro, and Gamba.

Critical conditional global well-posedness and scattering for cubic NLS in $\mathbb{R}^{3}$ - These notes are based on the work by Kenig and Merle.

Introduction to Fourier analysis on the torus -- These notes give an introduction to Fourier analysis on the torus $\mathbb{T}^{d}$. The notes include a treatment of Kolmogorov's construction of an $L^{1}(\mathbb{T})$ function whose Dirichlet means diverge pointwise almost everywhere.