arXiv subjects

Xiaoyutao Luo

Publications and source records attributed to Xiaoyutao Luo.

At least 19 recordsLinked to original sources

arXiv2026

Inviscid damping without derivatives near Couette flow

We study the long-time dynamics near Couette flow for the 2D Euler equations in unbounded geometries. We identify a mechanism of nonlinear inviscid damping at Yudovich regularity, distinct from classical phase mixing: velocity decay driven by spatial evacuation of vorticity. The mechanism leads to a geometry-sign classification of the dynamics. In the infinite channel, small nonnegative bounded-vorticity perturbations undergo global damping without derivative assumptions. In the whole plane, small nonnegative perturbations exhibit enhanced dispersion and damping along a set of times of density one, while non-positive perturbations remain confined and do not damp, even when arbitrarily small in Gevrey classes. Moreover, damping can coexist with infinite-time growth of vorticity derivatives, even under arbitrary shear modulation. The classification is obtained through an interplay between new Lyapunov functionals and Hamiltonian conservation.

math.AP↗

arXiv2026

Asymptotic stability of shear flows for 2D Euler equations at Yudovich regularity

The nonlinear asymptotic stability of shear flows in the 2D Euler equations has traditionally been linked to inviscid damping in the periodic setting. Since Gevrey regularity is required to suppress the ``echo'' phenomenon, asymptotic stability is known to be impossible in Sobolev spaces. In this paper, we identify a distinct stabilizing mechanism available in the infinite channel: the advection of vorticity to spatial infinity. We establish nonlinear asymptotic stability for the 2D Euler equations in the infinite channel $\mathbb{R}\times[0,1]$ at the minimal regularity of the Yudovich class ($L^{\infty}$ vorticity). Specifically, for a class of non-negative shear flows with a curvature bound, any $L^\infty$-small, compactly supported vorticity perturbation leads to decay on compact subsets and weak convergence to zero.

math.AP↗

arXiv2026

Norm Inflation For The Critical SQG Equation

We consider the critical dissipative surface quasi-geostrophic (SQG) equation on $\mathbb{R}^2$ or $\mathbb{T}^2$. Despite global regularity of the equation, we show that the data-to-solution map at the critical level $H^1$ is not uniformly bounded. We construct solutions that experience $H^1$ norm inflation from smooth, compactly supported initial data with large $H^1$ norm. We also demonstrate small-data norm inflation in supercritical Sobolev spaces $W^{β,p}$ for $1<p<2$ and $1\leβ<\tfrac{2}{p}$.

math.AP↗

arXiv2026

Flexibility and rigidity for the Couette flow in the infinite channel

We investigate the existence of stationary and traveling wave solutions to the 2D Euler equations near the Couette flow in the infinite channel $\mathbb{R} \times [-1,1]$. For Sobolev spaces $W^{s,p}$ or Hölder spaces $C^s$, we identify the index $s= 1+ \frac1p $ as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any $s<1+ \frac1p$ we prove the existence of $C^\infty$ smooth, compactly supported steady states and traveling waves arbitrarily close to the Couette flow in all $W^{s,p}$ and $C^{1-}$. Conversely, we establish the non-existence of such relative equilibria in $ W^{s,p}$ with $s>1+ \frac1p$ or $C^{1+}$. A notable feature of the variational construction is that these flexible solutions belong to every Gevrey class strictly below the analytic threshold.

math.AP↗

arXiv2025

Long time inviscid damping near Couette in Sobolev spaces

We give an elementary proof of long time inviscid damping for Sobolev perturbations near the Couette flow $(y,0)$ for the 2D Euler equations on $\mathbb{T} \times \mathbb{R}$. For any $s>1$ and any initial vorticity perturbation of size $O(ε)$ in $H^s$, we obtain velocity damping estimates up to a time scale $ t = O(ε^{-δ_s} )$, where $δ_s=1/3$ when $s\to 1+$ and $δ_s=1/2$ for $s>2$.

math.AP↗

arXiv2025

Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces

We prove that the incompressible Navier-Stokes equations exhibit norm inflation in $\dot B^{s}_{p,q}(\mathbb{R}^3)$ with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical $\dot B^{s}_{p,q} $ near the scaling-critical line $s = -1+ \frac{3}{p}$ except at $s=0$. The growth mechanism differs depending on the sign of the regularity index $s$: forward energy cascade driven by mixing for $s>0$ and backward energy cascade caused by un-mixing for $s<0$. The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations.

math.AP↗

arXiv2024

The $α$-SQG patch problem is illposed in $C^{2,β}$ and $W^{2,p}$

We consider the patch problem for the $α$-SQG system with the values $α=0$ and $α= \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint $C^{k,β}$ Hölder spaces, as well as in $W^{2,p},$ $1<p<\infty$ spaces. In stark contrast to the Euler case, we prove that for $0<α< \frac{1}{2}$, the $α$-SQG patch problem is strongly illposed in \emph{every} $C^{2,β} $ Hölder space with $β<1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for \emph{every} $W^{2,p}$ Sobolev space unless $p=2$.

math.AP↗

arXiv2024

Illposedness of incompressible fluids in supercritical Sobolev spaces

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0 < s < \frac{5}{2} $, we construct a $C^\infty_c$ initial velocity field with arbitrarily small $H^{s}$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equation for $0< s < \frac{1}{2} $, where $s = \frac{1}{2}$ is scaling critical for this equation.

math.AP↗

arXiv2024

Eulerian uniqueness of the $α$-SQG patch problem

We consider the patch problem of the $α$-SQG equation with $α=0$ being the 2D Euler and $α= \frac{1}{2}$ the SQG equations respectively. In the Eulerian setting, we prove the uniqueness of patch solutions of regularity $W^{2, \frac{1}{1-2α} +} $ when $0<α< \frac{1}{2}$ and $C^{1, 4α+ }$ when $0<α< \frac{1}{4} $. The proof is intrinsic to the modified Biot-Savart law and independent of the local existence of patch solutions.

math.AP↗

arXiv2023

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

We consider a family of singular surface quasi-geostrophic equations $$ \partial_{t}θ+u\cdot\nablaθ=-ν(-Δ)^{γ/2}θ+(-Δ)^{α/2}ξ,\qquad u=\nabla^{\perp}(-Δ)^{-1/2}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, where $ν\geq 0$, $γ\in [0,3/2)$, $α\in [0,1/4)$ and $ξ$ is a space-time white noise. For the first time, we establish the existence of infinitely many non-Gaussian $\bullet$ probabilistically strong solutions for every initial condition in $C^η$, $η>1/2$ $\bullet$ ergodic stationary solutions The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (M. Hairer, A theory of regularity structures). It also applies in the particular setting $α=γ/2$ which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato--Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.

math.PR↗

arXiv2023

Illposedness of $C^{2}$ vortex patches

It is well known that vortex patches are wellposed in $C^{1,α}$ if $0<α<1$. In this paper, we prove the illposedness of $C^{2}$ vortex patches. The setup is to consider the vortex patches in Sobolev spaces $W^{2,p}$ where the curvature of the boundary is $L^p$ integrable. In this setting, we show the persistence of $W^{2,p}$ regularity when $1 0$. The key ingredient is the evolution equation for the curvature, the dominant term in which turns out to be linear and dispersive.

math.AP↗

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.

math.AP↗

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 $d \geq 2$.

math.AP↗

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.

math.AP↗

arXiv2021

On nonexistence of splash singularities for the $α$-SQG patches

In this paper, we consider patch solutions to the $α$-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Strain \cite{GS}, providing a condition on the growth of curvature of the patch necessary for the splash and an exponential in time lower bound on the distance between patches with bounded curvature.

math.AP↗

arXiv2021

Anomalous dissipation, anomalous work, and energy balance for smooth solutions of the Navier-Stokes equations

In this paper, we study the energy balance for a class of solutions of the Navier-Stokes equations with external forces in dimensions three and above. The solution and force are smooth on $(0,T)$ and the total dissipation and work of the force are both finite. We show that a possible failure of the energy balance stems from two effects. The first is the \emph{anomalous dissipation} of the solution, which has been studied in many contexts. The second is what we call the \emph{anomalous work} done by the force, a phenomenon that has not been analyzed before. There are numerous examples of solutions exhibiting \emph{anomalous work}, for which even a continuous energy profile does not rule out the anomalous dissipation, but only implies the balance of the strengths of these two effects, which we confirm in explicit constructions. More importantly, we show that there exist solutions exhibiting \emph{anomalous dissipation} with zero \emph{anomalous work}. Hence the violation of the energy balance results from the nonlinearity of the solution instead of artifacts of the force. Such examples exist in the class $u \in L_t^{3 } B^{\frac{1}{3} -}_{3,\infty}$ and $f \in L_t^{2-} H^{-1}$, which implies the sharpness of many existing conditions on the energy balance.

math.AP↗

arXiv2020

Nonuniqueness of weak solutions for the transport equation at critical space regularity

We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L^\infty_t L^p$ when $\frac{1}{p} + \frac{1}{q} \leq 1$. For such vector fields, we show that in the regime $\frac{1}{p} + \frac{1}{q} > 1$, weak solutions are not unique in the class $ L^1_t L^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

math.AP↗

arXiv2020

Stationary and discontinuous weak solutions of the Navier-Stokes equations

We prove that there exists a nontrivial finite energy periodic stationary weak solution to the 3D Navier-Stokes equations (NSE). The construction relies on a convex integration scheme utilizing new stationary building blocks designed specifically for the NSE. The constructed family of approximate stationary solutions is also used to prove the existence of weak solutions of the NSE with energy profiles discontinuous on a dense set of positive Lebesgue measure.

math.AP↗