Haoming Zhu | Search arXiv

arXiv2026

Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space

In this paper, we study the axisymmetric Hall magnetohydrodynamics where $u = u^re_r+u^ze_z$ and $B = B^θe_θ$. We construct small, smooth, compactly supported, and axisymmetric initial data $(u_{0},B_{0})$ and prove that there is a unique smooth axisymmetric solution with norm inflation in $H^{σ-1}(\mathbb R^3) \times H^σ(\mathbb R^3)$ for $1<σ<7/2$. Moreover, we study the case with fractional dissipation $(-Δ)^βu$ and $(-Δ)^αB$, modifying the initial data slightly and proving that the solution inflates in $H^{σ-1}(\mathbb R^3) \times H^σ(\mathbb R^3)$ for $1<σ<7/2-2\max\{α,β\}$ and $ 0 \le α, β< 5/4$.

math.AP↗

arXiv2026

Half Strong Ill-Posedness of $2 \frac{1}{2}$D Electron Magnetohydrodynamics with Fractional Resistivity

We study the $2\frac{1}{2}$D electron magnetohydrodynamics (MHD): the electron MHD system that has $3$D magnetic field but is independent of $z$-variable. We establish a "half" strong ill-posedness result in $2\frac{1}{2}$D electron MHD with fractional resistivity $(-Δ)^α$ in the supercritical Sobolev space $H^β\times H^{β-1}$ for $3<β<4-2α$. Specifically, we construct small initial data $(a_0,b_0)$ whose solution develops a norm inflation in $a$ but the norm of $b$ remains small.

math.AP↗

arXiv2025

xRWA: A Cross-Chain Framework for Interoperability of Real-World Assets

Real-World Assets (RWAs) have recently attracted increasing attention as a means of bridging traditional financial instruments with decentralized infrastructures. By representing assets such as bonds, commodities, and real estate on blockchains, RWAs can enhance liquidity, broaden accessibility, and extend the scope of decentralized finance. Industry forecasts further suggest rapid growth of tokenized RWAs in the coming years, underscoring their potential role in the evolution of digital financial markets. However, when deployed across multiple blockchains, RWAs face challenges such as repeated authentication on different chains and inefficiency caused by multi-step settlement protocols. To address these issues, we present a cross-chain framework for RWAs that emphasizes identity management, authentication, and interaction. The framework integrates Decentralized Identifiers and Verifiable Credentials with customized attributes to support decentralized identification, and incorporates an authentication protocol based on Simplified Payment Verification to avoid redundant verification across chains. Furthermore, we design a cross-chain channel that enables the settlement of RWAs without requiring channel closure, thereby improving operational efficiency. We implement the framework and evaluate it through simulations, which confirm its feasibility and demonstrate improvements in efficiency for RWAs in cross-chain settings.

cs.CR↗