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Qirui Peng

Publications and source records attributed to Qirui Peng.

14 recordsLinked to original sources
arXiv2026

Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schr\"odinger equation

We construct, via convex integration, a nonzero singular weak solution of the one-dimensional periodic cubic nonlinear Schr\"odinger equation that is compactly supported in time, has zero initial datum, and satisfies $u\in\bigcap_{\alpha<1/6} C_t^0 H_x^\alpha \cap \bigcap_{1\leq p<3} C_t^0 L_x^p$. The construction adapts the intermittent slab building block of Gismondi, Ma, Pathak, and Radu. For the full cubic NLS, the spatially constant component generated by the nonlinearity cannot be discarded. We overcome this zero-output obstruction with a two-carrier perturbation: its unique zero-carrier, zero-slab-output interaction reproduces the full old error, including its zero Fourier mode, while the remaining interactions are perturbative in a negative Wiener norm. The nonlinear term is defined through a signed absolute-Fourier summability condition and is the common limit generated by every admissible Fourier cutoff. Whenever a singular solution belongs to $C_t^0 L_x^3$, this nonlinear term agrees with the ordinary product $|u|^2u$. Since $H^{1/6}(\mathbb T)\hookrightarrow L^3(\mathbb T)$ and unconditional uniqueness holds in $C_t^0 H_x^{1/6}$ by a result of Guo, Kwon, and Oh, the threshold $1/6$ is sharp within the singular solution class considered here.

math.AP↗
arXiv2026

Current-Sheet Formation in Electron Magnetohydrodynamics with Split Fractional Dissipation

Thin current sheets are central small-scale structures in electron magnetohydrodynamics (EMHD), closely associated with energy dissipation and fast magnetic reconnection at electron scales. We study their formation numerically in a $2\frac{1}{2}$-dimensional EMHD system on a periodic domain with split fractional dissipation, where the magnetic potential and the vertical magnetic component are damped separately. The local theory is governed by a symmetric combined damping balance, but the numerical onset of small-scale growth need not follow this symmetry. A scaling analysis identifies the out-of-plane current as the primary concentration observable, since it is regularized only through the magnetic-potential equation. Using a validated Fourier pseudospectral exponential time-differencing solver with resolution-controlled diagnostics, we find a clear decay/concentration dichotomy. The onset boundary is markedly asymmetric: current-sheet formation appears to be controlled mainly by damping of the magnetic potential, rather than by the combined damping strength. The analyticity strip collapses to the grid scale, the concentration sharpens under grid refinement, and the observed growth is consistent with an energy-critical self-similar rate, with exponent near three. These experiments indicate that magnetic-potential damping is the apparent binding constraint for current-sheet concentration, refining the symmetric sum picture.

physics.plasm-ph↗
arXiv2026

Onsager-Type Energy Equality and Prodi--Serrin Uniqueness for Nernst--Planck Fluid Systems

We study weak solutions of electrodiffusion systems coupling the Nernst--Planck equations with fluid models. First, for the three-dimensional Nernst--Planck--Euler system, we establish an Onsager-type criterion for the validity of the coupled kinetic-electrostatic energy balance. The energy equality is shown to hold for weak solutions whose velocity satisfies critical Besov regularity and a vanishing dyadic flux condition. Furthermore, assuming the corresponding Onsager-type regularity for the ionic concentrations, we also prove parabolic regularity, preservation of non-negativity of the concentrations, and the associated charge-density energy identity. Second, for the three-dimensional Nernst--Planck--Navier--Stokes system, we prove a Prodi--Serrin-type uniqueness criterion for Leray--Hopf solutions: uniqueness in the Leray--Hopf class holds whenever the velocity field lies in the Ladyzhenskaya--Prodi--Serrin class $L^p_tL^q_x$ with $2/p+3/q=1$ and $q>3$. These results extend energy-equality and weak--strong uniqueness principles from incompressible fluid dynamics to electrodiffusion models involving convection, diffusion, and self-consistent electrostatic forcing.

math.AP↗
arXiv2026

A Gradient Recovery Method for Electron Magnetohydrodynamics with Fractional Dissipation

We propose and analyze a structure-preserving numerical method for the $2\tfrac{1}{2}$-dimensional (2.5D) electron magnetohydrodynamics system with fractional dissipation on the periodic torus. The method works directly with the magnetic field components and combines this component formulation with the gradient recovery operator of [T. Chu, H. Guo, and Z. Zhang, SIAM J. Numer. Anal., 63 (2025), pp. 23--53]. We establish discrete energy stability for a semi-implicit structure-preserving formulation and use an explicit-Hall integrating-factor implementation for efficient computation on periodic grids. The fractional dissipation is treated exactly in Fourier space, and the in-plane divergence constraint is enforced by a spectral Hodge projection. Numerical experiments demonstrate second-order spatial convergence and stable Hall-driven dynamics across several benchmark tests.

math.NA↗
arXiv2026

Local well-posedness for the two-and-a-half-dimensional EMHD system with split fractional dissipation

We study the $2\frac12$-dimensional electron magnetohydrodynamics (EMHD) system on $\mathbb T^2$ with componentwise fractional dissipation: $\partial_t a+a_yb_x-a_xb_y=-\Lambda^\alpha a$ and $\partial_t b-a_y\Delta a_x+a_x\Delta a_y=-\Lambda^\beta b$, where $0<\alpha,\beta<2$. This system is a $2\frac12$-dimensional reduction of the magnetic equation in Hall--MHD/EMHD under the ansatz $B=\nabla\times(ae_z)+be_z$. We prove local well-posedness for initial data $(a_0,b_0)\in H^{s+1}(\mathbb T^2)\times H^s(\mathbb T^2)$ with $s\geq 2-\varepsilon$, provided that $\alpha+\beta>2$. Thus neither component is required to carry a full Laplacian dissipation; the smoothing effects of the two fractional dissipations can be combined to control the Hall nonlinearity. The proof is based on Littlewood--Paley energy estimates, commutator bounds, and cancellations between the leading low--high interactions.

math.AP↗
arXiv2026

The Three-Dimensional Stochastic EMHD System: Local Well-Posedness and Maximal Pathwise Solutions

We study the three-dimensional stochastic electron magnetohydrodynamics (EMHD) system with fractional dissipation on the torus, driven by Stratonovich transport noise acting through divergence-free first-order operators. The noise generates an It\^o correction while preserving the transport structure of the Hall nonlinearity. Since the Hall term contains one more derivative, in the stochastic setting it must be controlled together with commutators arising from the transport operators. We develop a high-order Sobolev energy method based on Littlewood--Paley analysis and refined commutator estimates, which yields uniform bounds for Galerkin approximations in $H^s$ with $s > \tfrac{5}{2}$ together with suitable time regularity. Using stochastic compactness and identification of limits, we construct martingale solutions for initial data in $L^2(\Omega; H^s)$. Pathwise uniqueness follows from cancellations in the Hall term combined with a stochastic Gr\"onwall argument. An application of a Yamada--Watanabe type result then yields local pathwise well-posedness and the existence of maximal pathwise solutions.

math.PR↗
arXiv2025

Anomalous Dissipation at Onsager-Critical Regularity

We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the \(2\frac{1}{2}\)-dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.

math.AP↗
arXiv2025

Well-posedness of the relaxed Electron MHD equations with random diffusion

We investigate a three-dimensional active-vector relaxation of the electron magnetohydrodynamics (EMHD) equations without resistivity. The relaxation replaces the electron current by a generalized current defined through a fractional power of the Laplacian. We drive the system by multiplicative pseudo-differential noise and apply an exponential transformation that removes the stochastic differential term and produces an effective fractional damping. For deterministic divergence-free, mean-zero initial data, we prove local well-posedness up to a stopping time in suitable Gevrey spaces. Under a smallness condition, we further establish global well-posedness with high explicit probability.

math.AP↗
arXiv2024

Three dimensional stationary solutions of the Electron MHD equations

The goal of this paper is to construct non-trivial steady-state weak solutions of the three dimensional Electron Magnetohydrodynamics equations in the class of $H^s(\mathbb T^3)$ for some small $s > 0$. By exploiting the formulation of the stationary EMHD equations one can treat them as generalized Navier-Stokes equations with half Laplacian. Therefore with convex integration scheme we obtained such stationary weak solutions, which is not yet realizable in the case of classical 3D Navier-Stokes equations.

math.AP↗
arXiv2023

Non-unique weak solutions of forced SQG

We construct non-unique weak solutions $\theta\in C_t^0C_x^{0-}$ for forced surface quasi-geostrophic (SQG) equation. This is achieved through a convex integration scheme adapted to the sum-difference system of two distinct solutions. Without external forcing, non-unique weak solutions $\theta$ in space $C_t^0C_x^{\alpha}$ with $\alpha<-\frac15$ were constructed by Buckmaster, Shkoller and Vicol, and Isett and Ma.

math.AP↗
arXiv2023

Uniqueness for a stochastic ideal dyadic MHD model

We study a stochastic dyadic model with both forward and backward energy cascade mechanisms for the inviscid and non-resistive magnetohydrodynamics. For a particular class of stochastic forcing, we show weak uniqueness for the stochastic system. However the solution dissipates the energy which is formally an invariant quantity for the system.

math.AP↗
arXiv2023

Non-unique stationary solutions of forced SQG

We show the existence of non-unique stationary weak solutions for forced surface quasi-geostrophic (SQG) equation via a convex integration scheme. The scheme is implemented for the sum-difference system of two distinct solutions. Through this scheme, one observes the external forcing is naturally generated accompanying the flexibility in means of lack of uniqueness. It thus provides a transparent way to reveal the flexibility of the system with the presence of a forcing.

math.AP↗
arXiv2021

Kolmogorov's dissipation number and determining wavenumber for dyadic models

We study some dyadic models for incompressible magnetohydrodynamics and Navier-Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov's dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov's dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov's dissipation wavenumber.

physics.flu-dyn↗