arXiv subjects

Guoxu Yang

Publications and source records attributed to Guoxu Yang.

10 recordsLinked to original sources

arXiv2026

Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows

We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier--Stokes equations. More precisely, every smooth solution on $\mathbb R^5\setminus\{0\}$ satisfying \[ \sup_{R>0}R^{-2}\int_{B_R}|u|^3\,dx<\infty \] is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.

math.AP↗

arXiv2026

New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations

The Liouville problem for the three-dimensional stationary Navier--Stokes equations remains open, even for axisymmetric $D$-solutions. In this paper, we obtain two results based on decay in the cylindrical radial variable $r=|x'|$. (i). Using a new pointwise Calder\'on--Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo--Pan--Zhang (2020, JFA) and prove \[ |\nabla u_r|+|\nabla u_z| \lesssim r^{-5/4}[\log(\mathrm e+r)]^{5/4}, \quad |\omega_r|+|\omega_z| \lesssim r^{-9/8}[\log(\mathrm e+r)]^{9/8}, \quad r\gg1. \] (ii). We develop a new approach to Liouville theorems that improves the axisymmetric criteria of Wang (2019, JDE) and Zhao (2019, Nonlinear Anal.). Without any symmetry assumption, we show that a $D$-solution is trivial if one of the following holds: \[ (\mathrm a).\,\sup_{{|x'|=r,\, z\in\mathbb R}} |u(x',z)| \leq Cr^{-2/3}[\log(\mathrm e+r)]^{-\gamma}; \quad (\mathrm b).\, \sup_{{|x'|=r,\, z\in\mathbb R}} |\omega(x',z)| \leq Cr^{-5/3}[\log(\mathrm e+r)]^{-\gamma}, \] for $r\geq1$, where $\gamma>1/3$.

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arXiv2026

Global bounded solutions for a class of generalized Hillen-Painter models near Couette flow in $\mathbb{R}^2$

We investigate the global well-posedness of a class of generalized Hillen--Painter systems -- specifically, supercritical volume-filling chemotaxis models -- in $\mathbb{R}^2$ under the influence of Couette flow. It is well established that, in the absence of fluid flow, solutions to this system may develop finite-time singularities (blow-up) for arbitrary initial cell mass. It is proved that the introduction of a Couette flow with sufficiently large amplitude guarantees the global existence of solutions for all initial masses. By employing a novel frequency decomposition technique, we successfully remove the mass threshold limitation presented in previous studies on the domain $\mathbb{T}\times\mathbb{R}$ (Wang et al., Commun. Contemp. Math.), thereby establishing global regularity in the whole space without any smallness assumptions.

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arXiv2026

Blow-up suppression for the nematic liquid crystal flow via Couette flow on $\mathbb{R}^2$

As is well known, for the harmonic heat flow or liquid crystal flow in two-dimension, the solution may blow up when the initial energy is greater than $8\pi$. Motivated by Lai--Lin--Wang--Wei--Zhou (CPAM, 2022), where singular solutions were constructed in the presence of small-scale velocity fields, it is natural to ask whether large-scale velocities may play a stabilizing role, preventing the concentration of blow-up. Here we show that the blow-up phenomenon can be suppressed by a Couette flow whose amplitude is large enough under a weak assumption on the anisotropic norm of the initial data. In particular, we construct examples with initial energy exceeding $8\pi$ that satisfy our assumptions.

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arXiv2025

Quantitative blow-up suppression for the Patlak-Keller-Segel(-Navier-Stokes) system via Couette flow on $\mathbb{R}^2$

It is well known that solutions to the Patlak--Keller--Segel system on $\mathbb{R}^2$ blow up in finite time if the initial mass exceeds $8\pi$. In this paper, we investigate the mixing effect induced by a Couette flow $(Ay, 0)$ with a quantitatively determined amplitude $A$, which suppresses bacterial aggregation. For the Patlak--Keller--Segel system advected by such a flow on $\mathbb{R}^2$, we prove that the solutions remain global in time even for large initial mass, provided the amplitude $A$ is sufficiently large. Specifically, global well-posedness holds if $A$ satisfies a lower bound of the form $C_* \left(\| \langle D_x\rangle^{m} \langle {D_x}^{-1}\rangle^\epsilon n_{\mathrm{in}} \|_{L^2}^2+1\right)^{9/2}$. A notable feature of our result is the explicit estimate of the sufficient constant, given by $C_* = 2,058,614$. Furthermore, for the coupled Patlak--Keller--Segel--Navier--Stokes system near the Couette flow, we establish an analogous global existence result, provided the amplitude is sufficiently large in form of $C_*\|(n_{\rm in}, |D_x|^{1/3} n_{\rm in},\omega_{\rm in})\|_{Y_{m,\epsilon}}^9$.

math.AP↗

arXiv2025

Global bounded solutions of the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and logistic source

As is well-known, the solutions to the Patlak-Keller-Segel system in 3D may blow up in finite time regardless of any initial cell mass. In this paper, we investigate the existence of global bounded solutions of the 3D Patlak-Keller-Segel-Navier-Stokes system with a large initial cell mass via Couette flow or logistic source in a finite channel. On the one hand, it is proved that as long as the Couette flow is strong enough and the initial velocity is small, the bounded solutions of the system are global in time without any limitation on the initial mass $M$ if the logistic source term exists. On the other hand, if the logistic source term vanishes, it is proved that as long as the Couette flow is strong enough and the initial velocity is small, the solutions with a finite initial mass $M$, whose lower bound is $ \left(\frac{8\pi}{9}\right)^-$, are global in time.

math.AP↗

arXiv2025

Stability threshold of Couette flow for Boussinesq equations in $\mathbb{R}^2$

This paper establishes the asymptotic stability threshold for the Couette flow $(y,0)$ under the 2D Boussinesq system in $\mathbb{R}^2$. It was proved that for initial perturbations in Sobolev spaces with controlled low horizontal frequencies, the stability threshold is at most $\left\{\frac{1}{3}+, \frac{2}{3}+\right\}$, extending the known threshold results from the periodic case $\mathbb{T}_x \times \mathbb{R}_y$ to the whole space. The core innovations are twofold: First, the $\langle D_x^{-1} \rangle$ control on the initial data simultaneously resolves horizontal frequency singularities and optimizes integral indices when applying Young's convolution inequality. Second, we develop a modified multiplier $\mathcal{M}_3$ that effectively absorbs the $|D_x|^{1/3}$ derivative structure induced by the temperature equation while handling nonlinear echo cascades.

math.AP↗

arXiv2025

Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$

Motivated by the classification of solutions of harmonic functions, we investigate Liouville type theorems for the fractional Navier-Stokes equations in $\mathbb{R}^3$ under some conditions on the boundedness of fractional derivatives. We prove that the smooth solution must be a trivial solution provided that it uniformly converges to a nonzero constant vector at infinity by applying Lizorkin's multiplier theorem to establish $L^p$ estimates for the fractional linear Oseen system and Coifman-McIntosh-Meyer type commutator estimates for the dissipation term. It is noteworthy that the integrability of velocity is not required here.

math.AP↗

arXiv2025

Saint-Venant Estimates and Liouville-Type Theorems for the Stationary MHD Equation in $\mathbb{R}^3$

In this paper, we investigate a Liouville-type theorem for the MHD equations using Saint-Venant type estimates. We show that $ (u, B) $ is a trivial solution if the growth of the $ L^s $ mean oscillation of the potential functions for both the velocity and magnetic fields are controlled. Our growth assumption is weaker than those previously known for similar results. The main idea is to refine the Saint-Venant type estimates using the Froullani integral.

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arXiv2024

Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity

In this paper, we investigate Liouville type theorems for the three-dimensional steady-state MHD or Hall-MHD system under some asymptotic assumptions at infinity. Firstly, for the Hall-MHD system we obtain that $u$ and $B$ are constant vectors for any fluid viscosity, magnetic resistivity or Hall-coefficient when the magnetic field $B$ tends to a non-zero constant vector at infinity while the velocity field $u$ tends to $0$. Secondly, it also follows that $u$ and $B$ are constant for the Hall-MHD system when the velocity field tends to a constant vector at infinity while the magnetic field tends to $0$ without any assumptions on viscosity, magnetic resistivity or Hall-coefficient. One main difficulty lies in the Hall term, and we obtain the $L^p$ estimates of a generalized Oseen system with some supercritical terms via Lizorkin's theory and prove that the operator is stable by exploring Kato's stability theorem. Moreover, some similar results for the degenerate fluid viscosity or magnetic resistivity for the MHD system are also obtained, which is independent of interest.

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