About

Vlad Vicol is a faculty member at NYU Courant who has been elected to the American Academy of Arts and Sciences. His research focus includes areas related to mathematics, with notable recognition such as the election to the Academy alongside other distinguished faculty members. His contributions have been recognized through awards and honors, indicating a significant impact in his field. Specific details about his research interests, background, or key contributions are not provided in the page text.

Research topics

• Physics
• Mathematics
• Mechanics
• Mathematical analysis

Selected publications

• On putative self-similarity for incompressible 3D Euler

  arXiv (Cornell University) · 2026-02-19

  articleOpen accessSenior author

  We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent $γ$ which governs the rate of zooming in must be larger than $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that $γ\geq 1/2$. For axisymmetric solutions, we establish the bound $γ\geq 1/2$ in more general settings, including ones in which the outgoing property is not present.

  PublisherOA PDF

• On putative self-similarity for incompressible 3D Euler

  Open MIND · 2026-02-19

  preprintSenior author

  We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent $γ$ which governs the rate of zooming in must be larger than $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that $γ\geq 1/2$. For axisymmetric solutions, we establish the bound $γ\geq 1/2$ in more general settings, including ones in which the outgoing property is not present.

  DOI

• Finite time singularities in the Landau equation with very hard potentials

  Open MIND · 2026-02-05

  preprint

  We consider the inhomogeneous Landau equation with $γ\in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^α$-norm of the distribution function blows up for every $α>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.

  DOI

• Finite time singularities in the Landau equation with very hard potentials

  arXiv (Cornell University) · 2026-02-05

  articleOpen access

  We consider the inhomogeneous Landau equation with $γ\in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^α$-norm of the distribution function blows up for every $α>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.

  PublisherOA PDF

• Classical Euler flows generate the strong Guderley imploding shock wave

  ArXiv.org · 2025-10-22

  preprintOpen accessSenior author

  We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal--gas law ($γ>1$) arises from classical, radially symmetric, shock--free data. For such data prescribed at initial time $\mathrm{T_{in}} < 0$, we prove that the flow remains smooth up to a first singular time $t=\mathrm{T}_* \in (\mathrm{T_{in}}, 0)$, where a preshock forms with a $C^{1/3}$ cusp in the fast acoustic variable. From this preshock a unique, initially weak, regular shock is born, whose strength can be made arbitrarily large on a controlled time interval; the front then deforms onto the Guderley shock and implodes at the origin at the collapse time $t=0$. There exists a matching time $t=\mathrm{T_{fin}} \in (\mathrm{T}_*,0)$ such that on $[\mathrm{T_{fin}},0)$ the solution coincides exactly with the classical Guderley self--similar profile, and at $t=\mathrm{T_{fin}}$ the shock trajectory matches the self--similar front to all orders. As $t \to 0^-$, the Euler solution implodes at the center, and continues for $t>0$ as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.

  PublisherOA PDFDOI

• Gradient catastrophes and an infinite hierarchy of Hölder cusp‐singularities for 1D Euler

  Journal of the London Mathematical Society · 2025-08-01

  articleSenior author

  Abstract We establish an infinite hierarchy of finite‐time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with nonconstant entropy. Specifically, for all integers , we prove that there exist classical solutions, emanating from smooth, compressive, and nonvacuous initial data, which form cusp‐type gradient singularities in finite time, in which the gradient of the solution has precisely Hölder‐regularity. We show that such Euler solutions are codimension‐ stable in the Sobolev space .

  PublisherDOI

• A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy

  Communications in Analysis and Mechanics · 2025-01-01 · 2 citations

  articleOpen accessSenior author

  We provide a detailed analysis of the shock formation process for the non-isentropic 2d Euler equations in azimuthal symmetry. We prove that from an open set of smooth and generic initial data, solutions of the Euler equations form a first singularity or gradient blow-up. This first singularity is termed a Hölder $ C^{\frac{1}{3}} $ pre-shock, and our analysis provides the first detailed description of this cusp solution. The novelty of this work relative to [1] is that we herein consider a much larger class of initial data, allow for a non-constant initial entropy, allow for a non-trivial sub-dominant Riemann variable, and introduce a host of new identities to avoid apparent derivative loss due to entropy gradients. The method of proof is also new and robust, exploring the transversality of the three different characteristic families to transform space derivatives into time derivatives. Our main result provides a fractional series expansion of the Euler solution about the pre-shock, whose coefficients are computed from the initial data.

  PublisherDOI

• Anomalous diffusion via iterative quantitative homogenization: an overview of the main ideas

  ArXiv.org · 2025-03-14

  preprintOpen accessSenior author

  <div xmlns="http://www.tei-c.org/ns/1.0"> Anomalous diffusion is the fundamental ansatz of phenomenological theories of passive scalar turbulence, and has been confirmed numerically and experimentally to an extraordinary extent. The purpose of this survey is to discuss our recent result, in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. Our main contribution is to propose an analytical framework in which to study anomalous diffusion via a backward cascade of renormalized eddy viscosities. </div>

  PublisherOA PDF

• Anomalous Diffusion by Fractal Homogenization

  Annals of PDE · 2025-01-03 · 13 citations

  articleOpen accessSenior author
  PublisherOA PDFDOI

• Lecture Notes on Intermittent Weak Solutions of the Three-dimensional Navier-Stokes Equations

  Nečas center series · 2025-07-26

  book-chapterOpen accessSenior authorCorresponding
  PublisherOA PDFDOI

Recent grants

• CAREER: Nonlinear stability mechanisms and boundary layer singularities in fluid flows

  NSF · $357k · 2018–2023

• CAREER: Nonlinear stability mechanisms and boundary layer singularities in fluid flows

  NSF · $123k · 2017–2019

• Regularity, stability, and singular limits in fluid dynamics

  NSF · $104k · 2013–2015

• Mathematical Analysis of Fluid Flow at High Reynolds Number from the Point of View of Turbulence

  NSF · $117k · 2015–2018

• Regularity, stability, and singular limits in fluid dynamics

  NSF · $131k · 2012–2014

Frequent coauthors

• Tristan Buckmaster

  60 shared

• Igor Kukavica

  University of Southern California

  46 shared

• Peter Constantin

  Princeton University

  34 shared

• Steve Shkoller

  University of California, Davis

  29 shared

• Jacob Bedrossian

  22 shared

• Susan Friedlander

  18 shared

• Maria Colombo

  18 shared

• Nathan Glatt-Holtz

  Tulane University

  17 shared

Labs

• Applied Mathematics LaboratoryPI

Awards & honors

• NSF Collaborative Research Grant DMS-2307681
• Simons Investigators program