arXiv subjects

Siran Li

Publications and source records attributed to Siran Li.

At least 19 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

On well-posedness theory of very weak solutions to Navier-Stokes equations on irregular domains with nonhomogeneous Dirichlet boundary data

The well-posedness theory of very weak solutions is a central topic in mathematical hydrodynamics, especially in the regularity theory for Navier-Stokes equations. It has been fully developed for incompressible fluid flows on bounded domains in R^3 of C^{2,1}-regularity. In this paper, based on the analytic theories in [D. Breit and A. Gaudin, ArXiv Preprint: 2511.19091 (2025)] and [V.G. Maz'ya and T.O. Shaposhnikova, Vol.337, Grundlehren der mathematischen Wissenschaften (2009)], we establish the well-posedness theory of very weak solutions to the Navier-Stokes equations on bounded Lipschitz domains whose boundary has local graphing functions with sufficiently small Sobolev multiplier norm, which contain the bounded Lipschitz domains with sufficiently small Lipschitz constants as a special case.

math.AP↗

arXiv2026

Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form

This paper is concerned with the existence theory of isometric immersions of surfaces with negative Gaussian curvature into the three-dimensional Euclidean space. We reformulate the Gauss--Codazzi equations, i.e., the partial differential equations for isometric immersions, into hyperbolic conservation laws for the flows of Chaplygin gas with nonzero source terms. Then, employing theories of invariant regions and compensated compactness, we establish for any given $p \in [2,\infty[$ the existence of $W^{2,p}_{\rm loc}$-isometric immersions of various general families of metrics over infinite strips $\mathbb{R}\times [0,T]$ with arbitrarily large $T$. Such metrics include those of various classical minimal surfaces: helicoid, catenoid, and Enneper surfaces, as well as metrics in isothermal coordinates or of the ``reciprocal-type''. In our fluid dynamical formulation of the isometric immersion problem, we specialise in the case that the two Riemann invariants for the associated hyperbolic conservation law remain bounded and of distinctive signs, and obtain $L^p$-solutions to the initial-boundary value problem via the method of relative entropy with respect to a nonstationary ODE background. The isometric immersions constructed in this paper have second fundamental forms belonging to $L^p_{\rm loc}\setminus L^\infty_{\rm loc}$.

math.DG↗

arXiv2026

MATCHA: Matching Text via Contrastive Semantic Alignment

Reliable evaluation is essential for understanding large language model (LLM) performance, yet today's go-to metrics, namely token-overlap scores (e.g., ROUGE) and embedding-based measures (e.g., BERTScore), often misjudge semantic similarity of documents. Our study shows that both token-overlap metrics and embedding-based metrics routinely assign nearly identical scores to texts that directly contradict each other, thereby potentially masking fundamental errors. We introduce MATCHA, an automatic metric that jointly rewards semantic agreement with a reference and penalizes contradictions. MATCHA employs a dual-view perspective that measures (i) proximity to the gold text and (ii) distance from an adversarially generated counterfactual contradiction. In eight public benchmarks, MATCHA outperforms popular metrics, compared with human annotations on question-answering, image caption generation, natural language inference, summarization, and semantic textual similarity tasks. On the TruthfulQA dataset (i.e., a dataset without a training set, where no embedding-based metrics could locally train on), this improvement in terms of matching texts with a reference reaches 18.38% over ROUGE-L and 20.82% over BERTScore. Both quantitative comparison and qualitative human assessments confirm the efficacy and validity of MATCHA and uncover fundamental weaknesses in pre-existing metrics. Compared with 23 embedding models, including top state-of-the-art ones, used as a metric similar to BERTScore, MATCHA remains the most accurate in distinguishing correct from incorrect statements solely based on a reference. Our code and metric are publicly available (https://github.com/Siran-Li/MATCHA).

cs.CL↗

arXiv2026

Boundary epsilon regularity for incompressible Navier--Stokes equations via weak-strong uniqueness

We show that finite-energy weak solutions to the incompressible Navier--Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L^4_tL^4_x$-norm of the solution is smaller than a constant depending only on the domain. This answers a problem raised in [D. Albritton, T. Barker, and C. Prange, J. Math. Fluid Mech. 25 (2023), Paper No. 49]. Our proof relies on a new slicing construction near the boundary of the domain.

math.AP↗

arXiv2026

Inviscid Limit for Yudovich solution to heat conductive Boussinesq equation on two-dimensional periodic domain

We establish the inviscid limit of the Yudovich solution to the heat conductive Boussinesq equation with initial velocity and temperature/buoyancy in $L^2$ and initial vorticity in $L^\infty$ on the two-dimensional periodic domain ${\bf T}^2$. Given any finite time $T>0$ and $p \in [1,\infty[$, we show that the solution to the diffusive Boussinesq equation converges in $L^\infty(0,T; W^{1,p}({\bf T}^2))$ to the solution to the Euler--Boussinesq equation as the viscosity tends to zero, provided that the initial vorticity, velocity, and temperature/buoyancy converge strongly in $L^2$. Our proof adapts and extends the arguments in [P. Constantin, T. D. Drivas, and T. M. Elgindi, Comm. Pure Appl. Math. 75 (2022), 60--82] to forcing terms in $L^1(0,T; L^\infty({\bf T}^2))$.

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↗

arXiv2026

Questions beyond Pixels: Integrating Commonsense Knowledge in Visual Question Generation for Remote Sensing

With the rapid development of remote sensing image archives, asking questions about images has become an effective way of gathering specific information or performing semantic image retrieval. However, current automatically generated questions tend to be simplistic and template-based, which hinders the deployment of question answering or visual dialogue systems for real-world applications. To enrich and diversify the questions with both image content and commonsense knowledge, we propose a Knowledge-aware Remote Sensing Visual Question Generation model (KRSVQG). The proposed model incorporates related knowledge triplets from external knowledge sources to broaden the question content, while employing image captioning as an intermediary representation to ground questions to the corresponding images. Moreover, KRSVQG utilizes a vision-language pre-training and fine-tuning strategy, enabling the model's adaptation to low data regimes. To evaluate the proposed KRSVQG model, we construct two knowledge-aware remote sensing visual question generation datasets: the NWPU-300 dataset and the TextRS-300 dataset. Evaluations, including metrics and human assessment, demonstrate that KRSVQG outperforms existing methods and leads to rich questions, grounded in both image and domain knowledge. As a key practice in vision-language research, knowledge-aware visual question generation advances the understanding of image content beyond pixels, facilitating the development of knowledge-enriched vision-language systems with vision-grounded human commonsense.

cs.CV↗

arXiv2026

Knowledge-aware Visual Question Generation for Remote Sensing Images

With the rapid development of remote sensing image archives, asking questions about images has become an effective way of gathering specific information or performing image retrieval. However, automatically generated image-based questions tend to be simplistic and template-based, which hinders the real deployment of question answering or visual dialogue systems. To enrich and diversify the questions, we propose a knowledge-aware remote sensing visual question generation model, KRSVQG, that incorporates external knowledge related to the image content to improve the quality and contextual understanding of the generated questions. The model takes an image and a related knowledge triplet from external knowledge sources as inputs and leverages image captioning as an intermediary representation to enhance the image grounding of the generated questions. To assess the performance of KRSVQG, we utilized two datasets that we manually annotated: NWPU-300 and TextRS-300. Results on these two datasets demonstrate that KRSVQG outperforms existing methods and leads to knowledge-enriched questions, grounded in both image and domain knowledge.

cs.CV↗

arXiv2026

Global well-posedness for one-dimensional compressible Navier--Stokes system in dynamic combustion with small $BV\cap L^1$ initial data

We establish the global well-posedness theory of small BV weak solutions to a one-dimensional compressible Navier--Stokes model for reacting gas mixtures in dynamic combustion. The unknowns of the PDE system consist of the specific volume, velocity, temperature, and mass fraction of the reactant. For initial data that are small perturbations around the constant equilibrium state $(1, 0, 1, 0)$ in the $L^1(\mathbb{R}) \cap {\rm BV}(\mathbb{R})$-norm, we establish the local-in-time existence of weak solutions via an iterative scheme, show the stability and uniqueness of local weak solutions, and prove the global-in-time existence of solutions for initial data with small BV-norm via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This work is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier--Stokes and Navier--Stokes--Fourier systems developed in [T.-P. Liu, S.-H. Yu, Commun. Pure Appl. Math. 75 (2022), 223--348] and [H. Wang, S.-H. Yu, X. Zhang, Arch. Ration. Mech. Anal. 245 (2022), 375--477].

math.AP↗

arXiv2025

Benchmarking is Broken -- Don't Let AI be its Own Judge

The meteoric rise of AI, with its rapidly expanding market capitalization, presents both transformative opportunities and critical challenges. Chief among these is the urgent need for a new, unified paradigm for trustworthy evaluation, as current benchmarks increasingly reveal critical vulnerabilities. Issues like data contamination and selective reporting by model developers fuel hype, while inadequate data quality control can lead to biased evaluations that, even if unintentionally, may favor specific approaches. As a flood of participants enters the AI space, this "Wild West" of assessment makes distinguishing genuine progress from exaggerated claims exceptionally difficult. Such ambiguity blurs scientific signals and erodes public confidence, much as unchecked claims would destabilize financial markets reliant on credible oversight from agencies like Moody's. In high-stakes human examinations (e.g., SAT, GRE), substantial effort is devoted to ensuring fairness and credibility; why settle for less in evaluating AI, especially given its profound societal impact? This position paper argues that the current laissez-faire approach is unsustainable. We contend that true, sustainable AI advancement demands a paradigm shift: a unified, live, and quality-controlled benchmarking framework robust by construction, not by mere courtesy and goodwill. To this end, we dissect the systemic flaws undermining today's AI evaluation, distill the essential requirements for a new generation of assessments, and introduce PeerBench (with its prototype implementation at https://www.peerbench.ai/), a community-governed, proctored evaluation blueprint that embodies this paradigm through sealed execution, item banking with rolling renewal, and delayed transparency. Our goal is to pave the way for evaluations that can restore integrity and deliver genuinely trustworthy measures of AI progress.

cs.AI↗

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↗

arXiv2025

GarmageNet: A Multimodal Generative Framework for Sewing Pattern Design and Generic Garment Modeling

Realistic digital garment modeling remains a labor-intensive task due to the intricate process of translating 2D sewing patterns into high-fidelity, simulation-ready 3D garments. We introduce GarmageNet, a unified generative framework that automates the creation of 2D sewing patterns, the construction of sewing relationships, and the synthesis of 3D garment initializations compatible with physics-based simulation. Central to our approach is Garmage, a novel garment representation that encodes each panel as a structured geometry image, effectively bridging the semantic and geometric gap between 2D structural patterns and 3D garment geometries. Followed by GarmageNet, a latent diffusion transformer to synthesize panel-wise geometry images and GarmageJigsaw, a neural module for predicting point-to-point sewing connections along panel contours. To support training and evaluation, we build GarmageSet, a large-scale dataset comprising 14,801 professionally designed garments with detailed structural and style annotations. Our method demonstrates versatility and efficacy across multiple application scenarios, including scalable garment generation from multi-modal design concepts (text prompts, sketches, photographs), automatic modeling from raw flat sewing patterns, pattern recovery from unstructured point clouds, and progressive garment editing using conventional instructions, laying the foundation for fully automated, production-ready pipelines in digital fashion. Project page: https://style3d.github.io/garmagenet/.

cs.GR↗

arXiv2025

Spacetime decay of mild solutions and conditional quantitative transfer of regularity of the incompressible Navier--Stokes Equations from $\mathbb{R}^n$ to bounded domains

We are concerned with the "transfer of regularity" phenomenon for the incompressible Navier--Stokes Equations (NSE) in dimension $n \geq 3$; that is, the strong solutions of NSE on $\mathbb{R}^n$ can be nicely approximated by those on sufficiently large domains $\Omega \subset \mathbb{R}^n$ under the no-slip boundary condition. Based on the space-time decay estimates of mild solutions of NSE established by [On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac. 43 (2000);$L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985)] and others, we obtain quantitative estimates for the ``transfer of regularity'' on higher-order derivatives of velocity and pressure under the smallness assumptions of the Stokes' system and/or the initial velocity, thus complementing the results obtained by [Using periodic boundary conditions to approximate the Navier-Stokes equations on $\mathbb{R}^n$ and the transfer of regularity, Nonlinearity 34 (2021)] and [Quantitative transfer of regularity of the incompressible Navier-Stokes equations from $\Bbb R^3$ to the case of a bounded domain, J. Math. Fluid Mech. 23 (2021)].

math.AP↗

arXiv2025

Enhancing Retrieval-Augmented Generation: A Study of Best Practices

Retrieval-Augmented Generation (RAG) systems have recently shown remarkable advancements by integrating retrieval mechanisms into language models, enhancing their ability to produce more accurate and contextually relevant responses. However, the influence of various components and configurations within RAG systems remains underexplored. A comprehensive understanding of these elements is essential for tailoring RAG systems to complex retrieval tasks and ensuring optimal performance across diverse applications. In this paper, we develop several advanced RAG system designs that incorporate query expansion, various novel retrieval strategies, and a novel Contrastive In-Context Learning RAG. Our study systematically investigates key factors, including language model size, prompt design, document chunk size, knowledge base size, retrieval stride, query expansion techniques, Contrastive In-Context Learning knowledge bases, multilingual knowledge bases, and Focus Mode retrieving relevant context at sentence-level. Through extensive experimentation, we provide a detailed analysis of how these factors influence response quality. Our findings offer actionable insights for developing RAG systems, striking a balance between contextual richness and retrieval-generation efficiency, thereby paving the way for more adaptable and high-performing RAG frameworks in diverse real-world scenarios. Our code and implementation details are publicly available.

cs.CL↗

arXiv2024

On global existence and large-time behaviour of weak solutions to the compressible barotropic Navier--Stokes Equations on $\mathbb{T}^2$ with density-dependent bulk viscosity: beyond the Va\u{\i}gant--Kazhikhov regime

We are concerned with the compressible barotropic Navier--Stokes equations for a $\gamma$-law gas with density-dependent bulk viscosity coefficient $\lambda=\lambda(\rho)=\rho^\beta$ on the two-dimensional periodic domain $\mathbb{T}^2$. The global existence of weak solutions with initial density bounded away from zero and infinity for $\beta>3$, $\gamma>1$ has been established by Va\u{\i}gant--Kazhikhov [Sib. Math. J. 36 (1995), 1283--1316]. When $\gamma=\beta>3$, the large-time behaviour of the weak solutions and, in particular, the absence of formation of vacuum and concentration of density as $t \to \infty$, has been proved by Perepelitsa [\textit{SIAM J. Math. Anal.} 39 (2007/08), 1344--1365]. Huang--Li [J. Math. Pures Appl. 106 (2016), 123--154] extended these results by establishing the global existence of weak solutions and large-time behaviour under the assumptions $\beta >3/2$, $1< \gamma<4\beta-3$, and that the initial density stays away from infinity (but may contain vacuum). Improving upon the works listed above, we prove that in the regime of parameters as in Huang--Li, namely that $\beta >3/2$ and $1< \gamma<4\beta-3$, if the density has no vacuum or concentration at $t=0$, then it stays away from zero and infinity at all later time $t \in ]0,\infty[$. Moreover, assuming $\beta>1$, $\gamma>1$ and a technical condition, we establish the global existence of weak solutions on $\mathbb{T}^2$. One of the key ingredients of our proof is a novel application --- motivated by the recent work due to Danchin--Mucha [Comm. Pure Appl. Math. 76 (2023), 3437--3492] --- of Desjardins' logarithmic interpolation inequality.

math.AP↗

arXiv2024

A note for double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of Euler equations

We give an elementary proof for the interior double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded $C^2$-domain $\Omega \subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u \in C^{0,\gamma}(\Omega;\mathbb{R}^d)$ with some $0<\gamma<1/2$, we show that the pressure $p \in C^{0,2\gamma}_{\rm int}(\Omega)$. This is motivated by the studies of turbulence and anomalous dissipation in mathematical hydrodynamics and, recently, has been established in [L. De Rosa, M. Latocca, and G. Stefani, Int. Math. Res. Not. 2024.3 (2024), 2511--2560] over $C^{2,1}$-domains by means of pseudodifferential calculus. Our approach involves only standard elliptic PDE techniques, and relies on a variant of the modified pressure introduced in [C. W. Bardos, D. W. Boutros, and E. S. Titi, H\"{o}lder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains, Arch. Rational Mech. Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes]. The key novel ingredient of our proof is the introduction of two cutoff functions whose localisation parameters are carefully chosen as a power of the distance to $\partial\Omega$.

math.AP↗

arXiv2024

Some recent developments on isometric immersions via compensated compactness and gauge transforms

We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.

math.AP↗