Journal ArticleCommunications in Mathematical Physics · November 1, 2025
We develop a computer-assisted symbolic method to show that a linearized Boussinesq flow in self-similar coordinates gives rise to an invertible operator. ...
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Journal ArticleNonlinearity · July 1, 2025
We consider equations of the type: (Formula presented) , for general linear operators R in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localised solutions. Singularities can even form i ...
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Journal ArticleDuke Mathematical Journal · May 15, 2025
We consider the advection-diffusion equation on T2 with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale |log υ|, where υ is the diffusivity par ...
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Journal ArticleArchive for Rational Mechanics and Analysis · February 1, 2025
We consider steady states of the two-dimensional incompressible Euler equations on T2 and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions t ...
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Journal ArticleAnalysis and Pde · January 1, 2025
We prove strong ill-posedness in L∞ for linear perturbations of the 2-dimensional Euler equations of the form (Formula presented), where R is any nontrivial second-order Riesz transform. Namely, we prove that there exist smooth solutions that ar ...
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Journal ArticleArchive for Rational Mechanics and Analysis · December 1, 2024
We construct a divergence-free velocity field u:[0,T]×T2→R2 satisfying (Formula presented.) such that the corresponding drift-diffusion equation exhibits anomalous dissipation for all smooth initial data. We also show that, given any ...
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Journal ArticleInventiones Mathematicae · October 1, 2024
We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to establish a number ...
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Journal ArticleMathematische Annalen · December 1, 2023
We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocit ...
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Journal ArticleEMS Surveys in Mathematical Sciences · November 15, 2023
Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the ...
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Journal Article · April 11, 2023
We consider the advection-diffusion equation on $\mathbb{T}^2$ with a
Lipschitz and time-periodic velocity field that alternates between two
piecewise linear shear flows. We prove enhanced dissipation on the timescale
$|\log \nu|$, where $\nu$ is the diffu ...
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Journal ArticleMemoirs of the American Mathematical Society · March 1, 2023
The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m-fold symmetric vortex patches with corners ...
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Journal ArticleArchive for Rational Mechanics and Analysis · February 1, 2023
We study the behavior of solutions to the incompressible 2d Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier–Stokes problems. We exhibit a large family of ...
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Journal ArticlePhilosophical transactions. Series A, Mathematical, physical, and engineering sciences · June 2022
We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data [Formula: see text], [Formula: see text], we construct a divergence-free advecting velocity field [Formula: see text] (depe ...
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Journal ArticleArchive for Rational Mechanics and Analysis · March 1, 2022
We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible C∞([0 , T) × Td) ∩ L1([0 , T] ; C1 -(Td)) velocity field which ex ...
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Journal ArticleCommunications on Pure and Applied Mathematics · January 1, 2022
We prove that given initial data (Formula presented.), forcing (Formula presented.) and any T > 0, the solutions uν of Navier-Stokes converge strongly in (Formula presented.) for any p ∈ [1, ∞) to the unique Yudovich weak solution u of the Euler ...
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Journal ArticleAdvances in Mathematics · December 24, 2021
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: {(x1,x2,x3):0321}. In this domain, we prove l ...
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Journal ArticleNonlinearity · July 1, 2021
We introduce an active scalar equation with a similar structure to the 3D Euler equations. Through studying the behavior of scale-invariant solutions, we show that compactly supported Lipschitz solutions belonging to CR2 0 can become singular in finite tim ...
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