arXiv subjects

Changhe Yang

Publications and source records attributed to Changhe Yang.

3 recordsLinked to original sources

arXiv2025

Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation

The nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations is one of the central open problems in mathematical fluid dynamics. In this paper, we provide, to our knowledge, the first rigorous computer-assisted proof demonstrating such nonuniqueness. Inspired by earlier works in this area, we construct a Leray-Hopf solution in the self-similar setting and then establish the existence of a second solution by analyzing the stability of the linearized operator around this profile, showing that it corresponds to an unstable perturbation. To achieve this, we develop an innovative numerical method that computes candidate solutions with high precision and propose a framework for rigorously establishing exact solutions in a neighborhood of these candidates. A key step is to decompose the linearized operator into a coercive part plus a compact perturbation, which is further approximated by a finite-rank operator up to a small error. The invertibility of the linearized operator restricted to the image of this finite-rank approximation is then rigorously verified using computer-assisted proofs. This certifies the existence of an unstable eigenpair and, consequently, yields a second solution - indeed, infinitely many Leray-Hopf solutions.

math.AP↗

arXiv2023

Asymptotic analysis for Bloch electrons with Weyl nodes

In this paper, we study the semiclassical behavior of Bloch electrons in the presence of Weyl nodes, which are singular points in the band structure of certain materials. We carry out asymptotic analysis and present a rigorous derivation of the semiclassical asymptotic expansion of the current of Bloch electrons with the presence of Weyl nodes. The analysis shows that the current contains two parts, one independent of the Weyl nodes and the other a contribution from the singular points. This work provides a theoretical foundation towards a rigorous justification of recent scientific discoveries in Weyl semimetals. The main innovation of this paper is a new strategy to deal with the singular points with quantitative estimates, which may have broader applications in multiscale models with singularities.

math.AP↗

arXiv2023

The semiclassical limit from the Pauli-Poisswell/Darwin to the Euler-Poisswell/Darwin system by WKB methods

The self-consistent Pauli-Poisswell and Pauli-Darwin equations for 2-spinors are $O(1/c)$ (where $c$ denotes the speed of light) semi-relativistic approximations of the Dirac-Maxwell equation for 4-spinors coupled to the self-consistent electromagnetic fields generated by the charge and current densities of a fast moving electric charge. They consist of a vector-valued magnetic Schr\"odinger equation with the Stern-Gerlach term which couples spin and magnetic field, coupled to 1+3 Poisson equations as the magnetostatic approximation of Maxwell's equations. The Pauli-Poisswell and Pauli-Dariwn euqations are $O(1/c)$ models keeping both relativistic effects magnetism and spin, both of which are absent in the non-relativistic Schr\"odinger-Poisson equation and inconsistent in the magnetic Schr\"odinger-Maxwell equation. We prove the local in time semiclassical limit $\hbar \rightarrow 0$ to the Euler-Poisswell equation and Euler-Darwin equations based on WKB analysis and energy estimates. Moreover we obtain weak convergence of the monokinetic Wigner transform to the monokinetic scalar Wigner measure solving the Vlasov-Poisswell and Vlasov-Darwin equations and strong convergence of the macroscopic densities. We introduce the Euler-Poisswell/Darwin equation and prove local wellposedness and a blow up alternative.

math.AP↗