arXiv subjects

Thomas Y. Hou

Publications and source records attributed to Thomas Y. Hou.

At least 19 recordsLinked to original sources

arXiv2026

On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions

We study finite-time blowup for the parabolic--elliptic Keller--Segel system on $\mathbb{R}^d$ in dimensions $d\geq11$, where the problem is mass supercritical. For every integer $l\geq2$, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state $Q$ at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time $T$, \[ u(t,r)=\frac{1}{λ^2(t)}\left[Q\left(\frac{r}{λ(t)}\right) +ε\left(t,\frac{r}{λ(t)}\right)\right], \qquad λ(t)=c(T-t)^{\frac{l}{γ(d)}}(1+o(1)), \] where $c>0$ and $γ(d)=\frac12\bigl(d-2-\sqrt{(d-2)(d-10)}\bigr)$. The remainder $ε$ converges to zero in local $L^\infty$ norms and in a range of high-order homogeneous Sobolev norms. Since $2l>γ(d)$, the concentration scale is strictly smaller than the parabolic scale $\sqrt{T-t}$, and the resulting blowup is of type II. The $l$-th regime has exactly $l-1$ unstable radial modulation directions and is stable within a codimension-$(l-1)$ class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of $Q$, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller--Segel flow.

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arXiv2026

Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis

Inspired by numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite-time, nearly self-similar blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. The proof encounters several essential difficulties. One of the essential difficulties is to control a number of nonlocal terms that do not seem to offer any damping effect. Another essential difficulty is that the strong advection normal to the boundary introduces a large growth factor for the perturbation when using weighted $L^2$ or $H^k$ estimates. We overcome these difficulties by combining weighted $L^\infty$ estimates and weighted $C^{1/2}$ estimates, and by developing sharp functional inequalities using the symmetry properties of the kernels and some techniques from optimal transport. Moreover, we decompose the linearized operator into a leading order operator and a finite-rank operator. We design the leading order operator to obtain sharp stability estimates. The contribution from the finite-rank operator to linear stability is estimated by constructing approximate space-time solutions. These ingredients enable us to establish the nonlinear stability of the approximate self-similar profile and to prove stable nearly self-similar blowup.

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arXiv2026

Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous Numerics

This is Part II of our paper in which we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. In Part I of our paper \cite{ChenHou2023a}, we establish an analytic framework to prove nonlinear stability of an approximate self-similar blowup profile using a combination of weighted $L^\infty$ and weighted $C^{1/2}$ energy estimates. We reduce proving nonlinear stability to verifying several inequalities for the constants in the energy estimate which depend on the approximate steady state and the weights in the energy functional only. In Part II of our paper, we construct approximate space-time solutions with rigorous error control, which are used to obtain sharp stability estimates of the linearized operator in Part I. We also obtain sharp estimates of the regular part of the velocity using numerical integration with computer assistance. These results enable us to verify that the constants in the energy estimate obtained in Part I \cite{ChenHou2023a} indeed satisfy the inequalities for nonlinear stability. The nonlinear stability further implies the finite time singularity of the axisymmetric 3D Euler equations with smooth initial data and boundary.

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arXiv2026

A clarification on the distinction between nonlocal error and profile residual error in a computer-assisted proof of 3D Euler singularity

In \cite{zhang2025dimension}, the author discusses the applicability of a nonexistence result for self-similar profiles to the Hou--Luo scenario and raises questions about the approximate self-similar profile constructed in \cite{ChenHou2023a,ChenHou2023b}. In this note, we clarify two main points. First, the far-field discrepancy identified in \cite{zhang2025dimension} is the raw nonlocal Poisson error, not the profile residual error used in the stability proof of \cite{ChenHou2023a,ChenHou2023b}. The relevant nonlocal contribution to the profile residual contains the nonlocal velocity error multiplied by additional decaying factors, which provide crucial smallness in the far field estimates. Second, we examine the figure-based evidence used in \cite[Remark 2.4]{zhang2025dimension} to support the applicability of the assumption in \cite[Proposition 2.3(i)]{zhang2025dimension}, and show that the same grid-point data do not support that interpretation. Thus the comparisons and figure-based evidence in \cite{zhang2025dimension} neither invalidate the residual estimates in \cite{ChenHou2023a,ChenHou2023b} nor justify the claimed applicability of the angular-increase assumption to the Hou--Luo scenario.

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arXiv2026

Analytic finite-rank corrections for singularly weighted estimates in a computer-assisted proof of 3D Euler singularity

Computer-assisted proofs of self-similar singularity formation for fluid equations often rely on numerically constructed approximate profiles. One effective approach to establishing stability of perturbations around a numerically constructed profile is to perform weighted energy estimates with singular weights near the singularity. However, the weighted norms require exact local vanishing conditions that are not automatically preserved by the equations nor the numerical construction. In this paper, we review an analytic low-rank correction method first developed in [ChenHou2023a,ChenHou2023b] to overcome this difficulty. The numerical step determines coefficients, rigorous bounds, and low-order defect modes in explicit global basis representations, while the required vanishing conditions are enforced analytically through low-rank corrections derived from Taylor expansions of the relevant quantities represented in a smooth basis. For completeness, we briefly review the singularly weighted estimates and a quantitative finite-rank perturbation method in the 2D Boussinesq / 3D Euler stability argument, where singular weights and the required vanishing order arise. Against this background, we formulate the local correction principle in a simplified setting, explain the correction of the residual error in numerical constructions of approximate space-time solutions and the stream function, and discuss its broader applicability to computer-assisted stability analysis for nonlocal PDEs.

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arXiv2026

Axisymmetric type II blowup solutions to the three-dimensional Keller-Segel system

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blow up in finite time. In particular, the singularity is of type II, which locally admits a leading-order profile of the rescaled stationary solution of the two-dimensional system. Additionally, mass concentration occurs along a one-dimensional ring in the plane. In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate.

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arXiv2026

Exact Blowup Analysis for the Weak-Advection Hou--Li Model

We study self-similar singularity formation for the one-dimensional weak-advection Hou--Li model, a reduced model motivated by the axisymmetric Euler equations. In the periodic setting, we construct exact finite-time self-similar blowup solutions for $2/3<a<1$, with profiles that are neither focusing nor expanding. In the whole-space setting with a Neumann condition, we construct exact finite-time self-similar blowup solutions for the full range $0<a\leq1$, with profiles of focusing, non-expanding/non-focusing, or expanding form depending on the sign of the self-similar scaling parameter. The construction is based on a fixed-point formulation near the origin, followed by an ODE extension argument. We also establish regularity, asymptotic behavior, monotonicity properties of the profiles, and uniqueness up to the natural scaling invariance.

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arXiv2026

Forward Self-Similar Solutions to the 2D Hypodissipative Navier-Stokes Equations

We study the forward self-similar solutions to the $2$D hypodissipative Navier-Stokes equation with fractional diffusion $(-Δ)^α$ for $\frac{1}{2}<α<1$. We first show that for arbitrarily large $(1-2α)$-homogeneous initial data which are locally Lipschitz, there exists at least one weak solution whose profile differs from the self-similar profile of the fractional heat equation by an element of $H^α(\reall^2)$. Moreover, when $α\in(\frac{2}{3},1)$ we show that any such weak solution is actually smooth, hence a strong solution, and satisfies certain far field decay estimates. Finally, we provide numerical evidence for the nonuniqueness of the related $2$D Navier-Stokes equation with time-dependent viscosity.

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arXiv2026

Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy

We study the singularity formation mechanisms of the inviscid generalized Surface Quasi-Geostrophic (gSQG) equation on the whole space $\mathbb{R}^2$ and on the upper half-plane $\mathbb{R}^2_+$, allowing infinite energy. In each case, we derive a one-dimensional reduction that captures the leading-order singular behavior of the original 2D system, and use a fixed-point argument to show the existence of finite-time self-similar blow-up solutions for the 1D systems. We also perform numerical simulations for verification and visualization.

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arXiv2026

$L^2$-based stability of blowup with log correction for semilinear heat equation

We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around the approximate profile and we establish a weighted $H^k$ stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Consequently, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the $L^2$-based stability framework beyond exactly self-similar blowup and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting.

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arXiv2025

Nearly self-similar blowup of generalized axisymmetric Navier-Stokes equations

We numerically investigate the nearly self-similar blowup of the generalized axisymmetric Navier--Stokes equations. First, we rigorously derive the axisymmetric Navier--Stokes equations with swirl in both odd and even dimensions, marking the first such derivation for dimensions greater than three. Building on this, we generalize the equations to arbitrary positive real-valued dimensions, preserving many known properties of the 3D axisymmetric Navier--Stokes equations. To address scaling instability, we dynamically vary the space dimension to balance advection scaling along the r and z directions. A major contribution of this work is the development of a novel two-scale dynamic rescaling formulation, leveraging the dimension as an additional degree of freedom. This approach enables us to demonstrate a one-scale self-similar blowup with solution-dependent viscosity. Notably, the self-similar profile satisfies the axisymmetric Navier-Stokes equations with constant viscosity. We observe that the effective dimension is approximately 3.188 and appears to converge toward 3 as background viscosity diminishes. Furthermore, we introduce a rescaled Navier--Stokes model derived by dynamically rescaling the axial velocity in 3D. This model retains essential properties of 3D Navier-Stokes. Our numerical study shows that this rescaled Navier--Stokes model with two constant viscosity coefficients exhibits a nearly self-similar blowup with maximum vorticity growth on the order of O(10^{30}).

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arXiv2025

High precision PINNs in unbounded domains: application to singularity formulation in PDEs

We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is $4$ digits smaller than that obtained in \cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems.

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arXiv2025

KAN: Kolmogorov-Arnold Networks

Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability. For accuracy, much smaller KANs can achieve comparable or better accuracy than much larger MLPs in data fitting and PDE solving. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful collaborators helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today's deep learning models which rely heavily on MLPs.

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arXiv2025

On the expressiveness and spectral bias of KANs

Kolmogorov-Arnold Networks (KAN) \cite{liu2024kan} were very recently proposed as a potential alternative to the prevalent architectural backbone of many deep learning models, the multi-layer perceptron (MLP). KANs have seen success in various tasks of AI for science, with their empirical efficiency and accuracy demostrated in function regression, PDE solving, and many more scientific problems. In this article, we revisit the comparison of KANs and MLPs, with emphasis on a theoretical perspective. On the one hand, we compare the representation and approximation capabilities of KANs and MLPs. We establish that MLPs can be represented using KANs of a comparable size. This shows that the approximation and representation capabilities of KANs are at least as good as MLPs. Conversely, we show that KANs can be represented using MLPs, but that in this representation the number of parameters increases by a factor of the KAN grid size. This suggests that KANs with a large grid size may be more efficient than MLPs at approximating certain functions. On the other hand, from the perspective of learning and optimization, we study the spectral bias of KANs compared with MLPs. We demonstrate that KANs are less biased toward low frequencies than MLPs. We highlight that the multi-level learning feature specific to KANs, i.e. grid extension of splines, improves the learning process for high-frequency components. Detailed comparisons with different choices of depth, width, and grid sizes of KANs are made, shedding some light on how to choose the hyperparameters in practice.

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arXiv2024

On the stability of blowup solutions to the complex Ginzburg-Landau equation in R^d

Building upon the idea in \cite{HNWarXiv24}, we establish stability of the type-I blowup with log correction for the complex Ginzburg-Landau equation. In the amplitude-phase representation, a generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose normalization conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. It enables us to establish the full stability of the blowup by enforcing vanishing conditions via the choice of normalization and using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is captured by the energy estimates and refined estimates of the modulation parameters.

math.AP↗

arXiv2024

Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$

We provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with $C^α$ initial vorticity for a large range of $α$. We employ a highly effective adaptive mesh method to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution study shows that our numerical method is at least second-order accurate. Scaling analysis and the dynamic rescaling method are presented to quantitatively study the scaling properties of the potential singularity. We demonstrate that this potential blow-up is stable with respect to the perturbation of initial data. Our numerical study shows that the 3D axisymmetric Euler equations with our initial data develop finite-time blow-up when the Hölder exponent $α$ is smaller than some critical value $α^*$, which has the potential to be $1/3$. We also study the $n$-dimensional axisymmetric Euler equations with no swirl, and observe that the critical Hölder exponent $α^*$ is close to $1-\frac{2}{n}$. Compared with Elgindi's blow-up result in a similar setting \cite{elgindi2021finite}, our potential blow-up scenario has a different Hölder continuity property in the initial data and the scaling properties of the two initial data are also quite different. We also propose a relatively simple one-dimensional model and numerically verify its approximation to the $n$-dimensional axisymmetric Euler equations. This one-dimensional model sheds useful light to our understanding of the blow-up mechanism for the $n$-dimensional Euler equations.

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arXiv2024

Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$. Part II: the $N$-Dimensional Case

In Part II of this sequence to our previous paper for the 3-dimensional Euler equations \cite{zhang2022potential}, we investigate potential singularity of the $n$-diemnsional axisymmetric Euler equations with $C^α$ initial vorticity for a large range of $α$. We use the adaptive mesh method to solve the $n$-dimensional axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine the potential blow-up and capture its self-similar profile. Our study shows that the $n$-dimensional axisymmetric Euler equations with our initial data develop finite-time blow-up when the Hölder exponent $α<α^*$, and this upper bound $α^*$ can asymptotically approach $1-\frac{2}{n}$. Moreover, we introduce a stretching parameter $δ$ along the $z$-direction. Based on a few assumptions inspired by our numerical experiments, we obtain $α^*=1-\frac{2}{n}$ by studying the limiting case of $δ\rightarrow 0$. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the $n$-dimensional Euler equations. This one-dimensional model sheds useful light to our understanding of the blowup mechanism for the $n$-dimensional Euler equations. As shown in \cite{zhang2022potential}, the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in \cite{elgindi2021finite}.

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arXiv2024

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier-Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou-Li in \cite{hou2008dynamic}. This model is based on an approximation of the axisymmetric Navier-Stokes equations in the $r$ direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier-Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in $r$. This model shares many intrinsic properties similar to those of the 3D Euler and Navier-Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio \cite{de1990one, de1996partial} model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

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