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The α$$\alpha$$‐SQG patch problem is illposed in C2,β and W2,p

https://doi.org/10.1002/cpa.22236

Kiselev, Alexander; Luo, Xiaoyutao (April 2025, Communications on Pure and Applied Mathematics)

Abstract We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed ineveryHölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds foreverySobolev space unless .
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Free, publicly-accessible full text available April 1, 2026
Extreme temporal intermittency in the linear Sobolev transport: Almost smooth nonunique solutions

https://doi.org/10.2140/apde.2024.17.2161

Cheskidov, Alexey; Luo, Xiaoyutao (January 2024, Analysis & PDE)

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$$L^2$$-Critical Nonuniqueness for the 2D Navier-Stokes Equations

https://doi.org/10.1007/s40818-023-00154-9

Cheskidov, Alexey; Luo, Xiaoyutao (December 2023, Annals of PDE)

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Illposedness of $$C^{2}$$ Vortex Patches

https://doi.org/10.1007/s00205-023-01892-7

Kiselev, Alexander; Luo, Xiaoyutao (June 2023, Archive for Rational Mechanics and Analysis)

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On Nonexistence of Splash Singularities for the $$\alpha $$-SQG Patches

https://doi.org/10.1007/s00332-023-09893-2

Kiselev, Alexander; Luo, Xiaoyutao (April 2023, Journal of Nonlinear Science)

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Sharp nonuniqueness for the Navier–Stokes equations

https://doi.org/10.1007/s00222-022-01116-x

Cheskidov, Alexey; Luo, Xiaoyutao (May 2022, Inventiones mathematicae)

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Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity

https://doi.org/10.1007/s40818-020-00091-x

Cheskidov, Alexey; Luo, Xiaoyutao (June 2021, Annals of PDE)
null (Ed.)
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Anomalous Dissipation, Anomalous Work, and Energy Balance for the Navier--Stokes Equations

https://doi.org/10.1137/20M1354076

Cheskidov, Alexey; Luo, Xiaoyutao (January 2021, SIAM Journal on Mathematical Analysis)
null (Ed.)
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Energy equality for the Navier–Stokes equations in weak-in-time Onsager spaces

https://doi.org/10.1088/1361-6544/ab60d3

Cheskidov, Alexey; Luo, Xiaoyutao (April 2020, Nonlinearity)

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