Takashi Suzuki
We study Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation several profiles of blowup solution have been noticed. Here we show the residual vanishing.
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We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.
We study Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation several profiles of blowup solution have been noticed. Here we show collapse mass quantization with possible vanishing term.
On gelation for the Smoluchowski equation
本文重新审视既有证明，针对一大类齐次凝并核简洁地证明 Smoluchowski 方程弱解会在有限时间发生凝胶化与质量损失，并刻画 γ=1 时对数因子的作用。
High Order Vanishing Theorems for Nonsimple Blowup Solutions of Singular Liouville Equation
本文研究奇异 Liouville 方程中非简单爆破现象，证明其仅在系数函数导数趋于零时发生，确定了任意量子化奇异源的消失阶数，为消除非简单爆破情形和研究移动极点爆破解提供了理论工具。
Laplacian Vanishing Theorem for Quantized Singular Liouville Equation
本文针对带量化奇异源的奇异Liouville方程建立拉普拉斯消失定理，给出非简单爆破情形下系数函数拉普拉斯趋于零的结论，是该类方程的首个二阶估计，具有重要应用价值。
In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 Ψ+ (1 + Ψ)^P ΔΨ= 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $Ψ$ is initially near $0$ while $1 + Ψ$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + Ψ$ i…
On the stability of blowup solutions to the complex Ginzburg-Landau equation in R^d
本文基于动态重标度与调制参数方法，在无拓扑或谱分析下建立了复 Ginzburg-Landau 方程 I 型对数修正爆破解的完全稳定性。
For Gauss curvature equation (or more general Toda systems) defined on two dimensional spaces, the vanishing rate of certain curvature functions on blowup points is a key estimate for numerous applications. However, if these equations have singular sources, very few vanishing estimates can be found. In this article we consider a Toda system with singular sources defined on a Riemann surface and we prove a very surprising vanishing estimates and a reflection phenomenon for certain functions involving the Gauss curvature.
This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $\barμ>0,~2\barμ+n\barλ\geq0,~\barκ\geq 0$, and adiabatic exponent $γ>1$). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1…
Finite time blowup for Keller-Segel equation with logistic damping in three dimensions
本文研究带二次logistic阻尼的三维Keller-Segel方程，证明μ∈[0,1/3)时存在有限时间爆破解，通过构造次临界标度自相似近似解结合加权L²方法证明稳定性，给出新的I型爆破机制。
On Existence and Blowup Criterion of Strong Solutions to Cauchy Problem of 2D Full Compressible Navier-Stokes System with Vacuum
本文研究了带真空的二维完全可压缩 Navier-Stokes 系统 Cauchy 问题，利用 $A_p$ 权和奇性抵消技术获得了先验加权估计，进而建立了强解的局部存在性与唯一性，并给出了独立于温度的爆破准则。
In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}}…
We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström, and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of "flat" b…
Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
该研究拓展非线性色散PDE概率适定性理论至方差爆破外，以BBM方程为对象，通过两种重整化方法证明解的依分布收敛性，且结果适用于任意低正则高斯初值，还推广到分数阶噪声驱动情形。
We prove that the only global strong solution of the periodic rod equation vanishing in at least one point $(t_0,x_0)$ is the identically zero solution. Such conclusion holds provided the physical parameter $γ$ of the model (related to the finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa--Holm equation, corresponding to $γ=1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on th…
Physical blowups via buffered time change in a mean-field neural network
本文针对平均场整合-发放神经网络中的爆破奇异性问题，通过缓冲时间变换将物理爆破动力学定义为不动点问题的解，并证明在缓冲消失极限下可恢复唯一且全局定义的物理解，从而避免永恒爆破现象。
Co-existence of Type II blow-ups with multiple blow-up rates for five-dimensional heat equation with critical nonlinear boundary conditions
该研究针对带临界非线性边界条件的五维热方程，构造了在不同边界点处具有不同II型爆破速率的有限时间爆破解，首次证明了多重速率II型爆破的共存性，并发展了适配该类不稳定爆破的统一线性理论。
Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime
本文证明两个互补的小数据稳定性定理，分别给出球对称3D可压缩欧拉方程激波形成解的MGHD唯一存在性及带$1/r$密度尾部数据的全时空光滑全局存在性。
Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
本文通过理论、数值和机器学习方法研究偏微分方程中的奇点形成问题，提出新的分析框架和数值方法，提升对复杂方程中奇点形成机制的理解与预测能力。
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