arXiv subjects

Shumao Zhang

Publications and source records attributed to Shumao Zhang.

3 recordsLinked to original sources

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.

math.AP↗

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}.

math.AP↗

arXiv2021

Multiscale Invertible Generative Networks for High-Dimensional Bayesian Inference

We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior performance of MsIGN over previous approaches in posterior approximation and multiple mode capture. On the natural image synthesis task, MsIGN achieves superior performance in bits-per-dimension over baseline models and yields great interpret-ability of its neurons in intermediate layers.

stat.ML↗