arXiv subjects

Yuantu Zhu

Publications and source records attributed to Yuantu Zhu.

3 recordsLinked to original sources

arXiv2026

Existence of Weak Solutions to a Power-Law Model for Compressible Non-Newtonian Fluids on 1D Unbounded Domain

This paper is concerned with the analysis of a one-dimensional power-law model for compressible fluid dynamics on $\mathbb{R}$, in which the shear stress takes the form $\mu |\partial_{x}u|^{p-2}\partial_{x}u$, where $\mu$ is the viscosity coefficient and $u$ is the velocity. We prove that, in the singular limit $p\rightarrow\infty$, the solutions converge to functions $(\rho,u)$ satisfying $|\partial_{x}u|\leq 1$, $\tau = \pi \partial_{x}u$, $\pi \geq 0$, and $\pi (1 - |\partial_{x}u|) = 0$ a.e. on $\mathbb{R}$. Moreover, we rigorously justify the existence of weak solutions to the limiting equation. The convergence as $p \to \infty$ is obtained via domain truncation and compactness arguments, of which the key challenge is to show that the density remains bounded away from zero and infinity on any compact subset. This extends the recent result of Bresch, Burtea, and Szlenk [Nonlinearity 26 (2026), no. 5, Paper No. 055010.] from one-dimensional periodic domain to the whole real line.

math.AP↗

arXiv2026

A compensated compactness theorem for pseudodifferential operators on vector bundles

We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly $L^2_{\rm loc}$-convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a pseudo-differential operator $\mathscr{A}$ of order $s>0$ is precompact in $H^{-s}_{\rm loc}$, we show that a quadratic form $Q$ acting on this sequence converges in the distributional sense, provided that $Q$ vanishes on the operator cone of $\mathscr{A}$. This extends the classical Murat--Tartar theory of compensated compactness from constant-coefficient first-order differential constraints on Euclidean spaces to variable-coefficient pseudo-differential constraints of arbitrary order on semi-Riemannian manifolds.

math.FA↗

arXiv2025

SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although machine learning (ML)-based surrogate models have shown promise for efficiently approximating such dynamics, progress remains limited by the lack of a unified benchmark with controlled data generation and comprehensive evaluation. This gap is particularly significant for singular SPDEs, for which benchmark datasets are largely unavailable and reliable simulation requires numerically delicate schemes based on renormalization. Moreover, subtle differences in data-generation procedures, such as noise approximation, basis choice, and the inclusion of renormalization, can significantly affect the resulting datasets and, consequently, model evaluation. We introduce SPDEBench, the first unified benchmark for ML-based SPDE learning. SPDEBench provides ready-to-use datasets for physically and mathematically significant SPDEs on 1-3D domains with periodic or Dirichlet boundary condition. Both regular and singular SPDEs are taken into consideration. SPDEBench also incorporates representative ML baselines in operator learning, together with 7 evaluation metrics, including Sobolev and distributional metrics beyond the standard $L^2$-error. Supported by SPDEBench, we conduct systematic evaluations of model accuracy, robustness, and out-of-distribution generalization under controlled data variations. Our numerical results show that SPDE-aware architectures generally achieve stronger performance than generic operator-learning baselines. These findings establish SPDEBench as a reproducible and extensible resource, paving pathway for principled benchmarking and architecture design for stochastic spatio-temporal dynamics.

cs.LG↗