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Alexey Cheskidov

Publications and source records attributed to Alexey Cheskidov.

At least 19 recordsLinked to original sources
arXiv2026

Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation

We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\infty}_tL^\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.

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arXiv2026

Cascade mechanisms for Navier-Stokes blow-up

In a recent preprint (arXiv:2511.09556), we exhibited an inverse energy cascade for the 3D Navier-Stokes equations which results in "instantaneous" Type I blow-up and failure of uniqueness in certain sharp regularity classes. Our purpose here is to elucidate this phenomenon further in the setting of the Obukhov dyadic model, and to compare it to the better-known phenomenon of finite-time blow-up. When intermittency is low ($α\leq 2$), we recover our previous result from the Navier-Stokes setting; in the energy-supercritical case where intermittency is high ($α>2$), we show that the same can occur in the class of finite energy solutions. We present two different proofs: a soft approach using a Lyapunov function and an explicit multiscale construction. For the inviscid system we illustrate that a similar phenomenon occurs at lower regularity. Finally, we state a finite-time blow-up theorem for a mixed Desnyansky-Novikov-Obukhov model, in which a forward cascade from finitely supported data and no force produces a singularity that is of Type II only by a small margin, with features comparable to the recent forced Navier-Stokes blow-up of OpenAI.

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arXiv2026

Global boundedness, absorbing sets and mass persistene in two-dimensional chemotaxis-Navier-Stokes systems with weakly singular sensitivity and sub-logistic sources

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $Ω$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= Δn - χ\nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n - \frac{μn^2}{\log^η(n+e)}, c_t + u \cdot \nabla c &= Δc - αc + βn, u_t + u \cdot \nabla u &= Δu - \nabla P + n \nabla ϕ+ f, \nabla \cdot u &= 0, \end{cases} \end{equation*} where $r, μ, α, β, χ$ are positive parameters, $k, η\in (0,1)$, $ϕ\in W^{2,\infty}(Ω)$, and $f \in C^1\left(\barΩ\times [0, \infty)\right) \cap L^\infty\left(Ω\times (0, \infty)\right)$. We show that, under suitable conditions on the initial data and with no-flux/no-flux/Dirichlet boundary conditions, this system admits a globally bounded classical solution. Furthermore, the system possesses an absorbing set in the topology of $C^0(\barΩ) \times W^{1, \infty}(Ω) \times C^0(\barΩ; \mathbb{R}^2)$. Finally, we establish the persistence of the total mass of solutions, indicating that the population does not face extinction as a whole.

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arXiv2026

On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.

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arXiv2026

Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces

We study vortex stretching for the three-dimensional axisymmetric Euler equations without swirl in vorticity formulation. Danchin (2007) established global existence and uniqueness for bounded vorticity $ω_0$ provided $ω_0/r$ lies in the endpoint Lorentz space $L^{3,1}(\mathbb{R}^3)$ (together with a decay assumption on $ω_0$). We prove that this $L^{3,1}$ endpoint is sharp: for every Lorentz exponent $q>1$, we construct multi-ring data $ω_0 \in L^\infty (\mathbb{R}^3)$ with $ω_0/r\in L^{3,q}(\mathbb{R}^3)$ that produce $L^\infty$-norm inflation of the vorticity; moreover, within the same class, we obtain instantaneous blow-up from data with infinitely many rings. Our initial data are inspired by the Kim--Jeong dyadic ring superposition (2022), but we crucially generalize it by allowing flexible conical support geometry for the ring profile. In the regime where outer rings are dominant -- a multiscale viewpoint appearing in recent works including Kim--Jeong (2022) and Cordoba--Martinez-Zoroa--Zheng (2025) -- we obtain a forward-in-time ODE cascade for ring amplitudes and aspect ratios in which vortex stretching weakens its own future forcing: as a ring amplifies, incompressibility flattens it, the aspect ratio collapses, and the induced stretching coefficient is geometrically depleted. A key new ingredient is a profile-localization argument that freezes the relevant Biot--Savart kernel and makes this depletion explicit, enabling us to exploit a monotone "productive window" (controlled by the cone slope) together with an exact cascade identity. This propagates stretching across scales and gives a robust lower bound on cumulative stretching, yielding ill-posedness in the full range $q>1$.

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arXiv2026

Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb T^d\times([0,T]\setminus\{T_*\})$. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier--Stokes flow, which transfers energy from infinitely high to low frequencies. The result holds in all dimensions $d\geq2$.

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arXiv2025

Anomalous Dissipation at Onsager-Critical Regularity

We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the \(2\frac{1}{2}\)-dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.

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arXiv2025

Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space

We construct global smooth solutions to the incompressible Navier--Stokes equations in $\mathbb{R}^3$ for initial data in $L^2$ satisfying some smallness condition. The high-frequency part is assumed to be small in $BMO^{-1}$, while the low-frequency part is assumed to be small only in $\dot B^{-1}_{\infty,\infty}$. Since $BMO^{-1}$ is strictly embedded in $\dot B^{-1}_{\infty,\infty}$, our assumption is weaker than that of Koch and Tataru (2001), which we also demonstrate with an example of finite energy divergence-free initial data. Also, our solutions attain the initial data in the strong $L^2$ sense, and hence satisfy the energy balance for all time.

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arXiv2025

Global dissipative solutions of the 3D Naiver-Stokes and MHD equations

For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term.

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arXiv2024

Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations

The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with smooth energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong. Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction.

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arXiv2024

Dissipation anomaly and anomalous dissipation in incompressible fluid flows

Dissipation anomaly, a phenomenon predicted by Kolmogorov's theory of turbulence, is the persistence of a non-vanishing energy dissipation for solutions of the Navier-Stokes equations as the viscosity goes to zero. Anomalous dissipation, predicted by Onsager, is a failure of solutions of the limiting Euler equations to preserve the energy balance. Motivated by a recent dissipation anomaly construction for the 3D Navier-Stokes equations by Bruè and De Lellis (2023), we prove the existence of various scenarios in the limit of vanishing viscosity: the total and partial loss of the energy due to dissipation anomaly, absolutely continuous dissipation anomaly, anomalous dissipation without dissipation anomaly, and the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. We also discover a relation between dissipation anomaly and the discontinuity of the energy of the limit solution. Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias (2002) upper bound is sharp.

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arXiv2023

$L^2$-critical nonuniqueness for the 2D Navier-Stokes equations

In this paper, we consider the 2D incompressible Navier-Stokes equations on the torus. It is well known that for any $L^2$ divergence-free initial data, there exists a global smooth solution that is unique in the class of $C_t L^2$ weak solutions. We show that such uniqueness would fail in the class $C_t L^p$ if $ p<2$. The non-unique solutions we constructed are almost $L^2$-critical in the sense that $(i)$ they are uniformly continuous in $L^p$ for every $p<2$; $(ii)$ the kinetic energy agrees with any given smooth positive profile except on a set of arbitrarily small measure in time.

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arXiv2023

Dyadic models for fluid equations: a survey

Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years, including the groundbreaking work of Ladyzhenskaya on the Navier-Stokes equations. However crucial questions such as the existence, uniqueness and regularity of the three dimensional Navier-Stokes equations remain open. Partly because of this mathematical challenge and partly motivated by the phenomena of turbulence, insights into the full PDEs have been sought via the study of simpler approximating systems that retain some of the original nonlinear features. One such simpler system is an infinite dimensional coupled set of nonlinear ordinary differential equations referred to a dyadic model. In this survey we provide a brief overview of dyadic models and describe recent results. In particular, we discuss results for certain dyadic models in the context of existence, uniqueness and regularity of solutions.

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arXiv2022

Extreme temporal intermittency in the linear Sobolev transport: almost smooth nonunique solutions

In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in \mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions $d \geq 2$.

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arXiv2022

Sharp nonuniqueness for the Navier-Stokes equations

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension $d \geq 2$ and given any $ p<2$, we show the nonuniqueness of weak solutions in the class $L^{p}_t L^\infty$, which is sharp in view of the classical Ladyzhenskaya-Prodi-Serrin criteria. The proof is based on the construction of a class of non-Leray-Hopf weak solutions. More specifically, for any $ p<2$, $q<\infty$, and $\varepsilon>0$, we construct non-Leray-Hopf weak solutions $ u \in L^{p}_t L^\infty \cap L^1_t W^{1,q}$ that are smooth outside a set of singular times with Hausdorff dimension less than $\varepsilon$. As a byproduct, examples of anomalous dissipation in the class $L^{ {3}/{2} - \varepsilon}_t C^{ {1}/{3}} $ are given in both the viscous and inviscid case.

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arXiv2022

Volumetric theory of intermittency in fully developed turbulence

This study introduces a new family of volumetric flatness factors which give a rigorous parametric description of the phenomenon of intermittency in fully developed turbulent flows. These quantities gather information about the most "active" part of a velocity field at each scale $\ell$, and allows one to define a dimension function $p \to D_p$ that recovers intermittency correction to the structure exponents $ζ_p$ in an explicit way. In particular, the predictions of the Frisch-Parisi multifractal formalism can be recovered in a systematic and rigorous way. Within this framework we identify active regions that carry the most energetic part of a velocity field at a given scale $\ell$. A threshold for what constitutes to be active is defined explicitly. Active regions have proven to be experimentally observable in our previous joint work \cite{Ph-paper}, and shown to capture concentration of the energy cascade as $\ell \to 0$, in \cite{CS2014}. We present several examples of fields which exhibit arbitrary multifractal spectrum within theoretically permitted limitations. At the same time we demonstrate with the use of a probabilistic argument that a random field is expected to produce the classical K41 spectrum in the limit $\ell\to 0$. Intermittent deviations from K41 theory are estimated at any finite scale also. Lastly, we present a detailed information-theoretic analysis of the introduced objects. In particular, we quantify concentration of a given source-field in terms of the volume factors, thresholds, and active regions.

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