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Changyou Wang

Publications and source records attributed to Changyou Wang.

At least 19 recordsLinked to original sources
arXiv2026

On Variational Approximations For Wave Maps

n this paper, we revisit the existence of global weak solutions of wave maps from $\R^n$ into the sphere $\mathbb{S}^{L-1}$, $\Box u\perp T_u \mathbb{S}^{L-1}$, by establishing it as a singular limit of maps from $\R^n\times \R_+$ to $\mathbb S^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\varepsilon$, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi \cite{Giorgi1996}, which has been implemented by Serra and Tilli \cite{Serra-Tilli2012, Serra-Tilli2016} for certain class of nonlinear wave equations. This approach is also applicable to the $SO(m)$-target manifold.

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arXiv2026

Energy identity for stationary biharmonic mappings into spheres in supercritical dimensions

Energy identity for harmonic type maps in supercritical dimensions is an important and difficult problem. For sphere-valued harmonic maps, the first breakthrough was achieved by Lin-Rivi\`ere [Duke Math. J. 2002]. In this paper, by adapting their strategy, we establish the energy identity for stationary biharmonic maps into spheres in supercritical dimensions $n\ge 5$.

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arXiv2026

Quantitative stratification and global regularity for 1/2-harmonic mappings

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.

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arXiv2026

Remarks on the heat flow of harmonic maps into CAT(0)-spaces

In this paper, we present an alternate, elementary proof of the local Lipschitz regularity of the suitable weak solution of heat flow of harmonic maps into CAT(0)-metric spaces, whose existence was established by Lin, Segatti, Sire, and Wang through an elliptic regularization approach. The ideas of the proof are inspired by Korevaar and Schoen, and they work for any CAT(0)-metric space $(X,d)$ as the target and any complete Riemanan manifold $(M,g)$, with positive injectivity radius and bounded curvature, as the domain.

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arXiv2026

On a Problem Posed by Brezis and Mironescu

The purpose of this note is to present a positive answer to an open problem proposed in the recent book \cite{Brezis-Mironescu} by H. Brezis and P. Mironescu. It has been stated in this book {\it Sobolev Maps to the Circle} as Proposition 4.3. We demonstrate, in particular, the value of the least mass of the area minimizing integral rectifiable currents with a given boundary equals to the infimum of areas among smoothly immersed submanifolds with the same boundary, under the assumption that the boundary is that of a smooth submanifold.

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arXiv2026

Heat flow of harmonic maps into CAT($0$)-spaces

We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables for such solutions into any CAT$(0)$-space, answering a long-standing open problem in the field. Our approach is based on an elliptic regularization of the gradient flow of the Dirichlet energy and even in the case of smooth Riemannian targets provides a novel viewpoint, together with a new Dynamical Variational Principle and a new proof of the celebrated Eells-Sampson theorem. The spatial Lipschitz regularity for such weak solutions is achieved by fully exploiting the variational structure of the problem at the regularized level and introducing a parabolic frequency function of Almgren-Poon type. Our contribution is the first instance of the use of monotonicity methods for parabolic deformations of maps into singular targets.

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arXiv2025

Eigenframe discontinuities of the Q-tensor model

In this paper, we study the defect structure of minimizer of a Landau-de Gennes energy functional in three-dimensional domains, subject to constraint $|Q|=1$. The set of defects is identified by discontinuities in both the eigenframe and the leading eigenvector. Through a blow-up analysis, we prove that the defect set is 1-rectifiable and classify the asymptotic profile of the leading eigenvector near singularities. This generalizes some previous results on the structure of ring disclinations in the $Q$-tensor model.

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arXiv2025

The Lamm-Rivi\`ere system II: energy identity

In this paper, we establish an angular energy quantization for the following fourth order inhomogeneous Lamm-Rivi\`ere system $$ \Delta^2u=\Delta(V\cdot\nabla u)+\text{div}(w\nabla u)+W\cdot\nabla u+f $$ in dimension four, with an inhomogeneous term $f\in L\log L$.

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arXiv2025

On rigidity of the steady Ericksen-Leslie system

We study solutions, with scaling-invariant bounds, to the steady simplified Ericksen-Leslie system in $\mathbb{R}^n\setminus \{0\}$. When $n=2$, we construct and classify a class of self-similar solutions. When $n\ge 3$, we establish the rigidity asserting that if $(u,d)$ satisfies a scaling-invariant bound with a small constant, then $u\equiv 0$ and $d=$ constant for $n\geq 4$ or $u$ is a Landau solution and $d=$ constant for $n=3$. Such a smallness condition can be weaken when $n=4$ or the solutions are self-similar.

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arXiv2024

On forward self-similar heat flow of harmonic maps

For any $k$-dimensional smooth, compact Riemannian manifold $(N, h)\subset\mathbb R^L$ without boundary, there exists an $\varepsilon_0>0$ such that for any homogeneous of degree zero map $u_0(x)=\phi_0(\frac{x}{|x|}):\mathbb R^n\to N$ ($n\ge 2$), if $\|\nabla\phi_0\|_{L^n(\mathbb S^{n-1})}\le\varepsilon_0$ then there is a unique solution $u:\mathbb R^n\times (0,\infty)\to N$ to the heat flow of harmonic map \eqref{HF1} and \eqref{IC}, which is forward self-similar and belongs to $C^\infty(\R^n\times (0,\infty))\cap C^{\frac1{n}}(\R^n\times [0,\infty)\setminus \{(0,0)\})$.

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arXiv2024

Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system

In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter $\varepsilon$ tends to zero. The proof relies on the one hand on the use of a blow-up argument to rule out energy concentration off the $z$-axis, which exploits the topological restrictions of the axisymmetry. On the other hand, possible limiting energy concentrations on the $z$-axis can be dealt by a cancellation argument at the origin. Once more, the axisymmetry plays a substantial role. We will also show that the set of axisymmetric solutions without swirl $(u,d)$ to the simplified Ericksen-Leslie system is compact under weak convergence in $L^\infty_tL^2_x\times L^2_tH^1_x$.

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arXiv2023

A decoupled structure preserving scheme for the Poisson-Nernst-Planck Navier-Stokes equations and its error analysis

We consider in this paper a numerical approximation of Poisson-Nernst-Planck-Navier- Stokes (PNP-NS) system. We construct a decoupled semi-discrete and fully discrete scheme that enjoys the properties of positivity preserving, mass conserving, and unconditionally energy stability. Then, we establish the well-posedness and regularity of the initial and (periodic) boundary value problem of the PNP-NS system under suitable assumptions on the initial data, and carry out a rigorous convergence analysis for the fully discretized scheme. We also present some numerical results to validate the positivity-preserving property and the accuracy of our scheme.

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arXiv2022

On a variational problem of nematic liquid crystal droplets

Let $μ>0$ be a fixed constant, and we prove that minimizers to the following energy functional \begin{align*} E_f(u,Ω):=\int_Ω|\nabla u|^2+μP(Ω) \end{align*}exist among pairs $(Ω,u)$ such that $Ω$ is an $M$-uniform domain with finite perimeter and fixed volume, and $u \in H^1(Ω,\mathbb{S}^2)$ with $u =ν_Ω$, the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of $Ω$. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by \begin{align*} E_f(u,Ω):=\int_Ω |\nabla u(x)|^2 \,dx + \int_{\partial^* Ω} f\big(u(x)\cdot ν_Ω(x)\big) \,d\mathcal{H}^2(x), \end{align*}where $\partial^* Ω$ is the reduced boundary of $Ω$ and $f$ is a convex positive function on $\mathbb R$. We prove that minimizers of $E_f$ also exist among $M$-uniform outer-minimizing domains $Ω$ with fixed volume and $u \in H^1(Ω,\mathbb{S}^2)$.

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arXiv2022

Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.

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arXiv2021

$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

We establish an optimal $L^p$-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $n\ge 5$: $$ \Delta^2 u=\Delta(D\cdot\nabla u)+div(E\cdot\nabla u)+(\Delta\Omega+G)\cdot\nabla u +f \qquad \ {\rm{in}}\ B^n, $$ where $\Omega\in W^{1,2}(B^n, so_m)$ is antisymmetric and $f\in L^p(B^n)$, and $D, E, \Omega, G$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $\nabla u$ and $\nabla^2 u$. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Rivi\`ere, Struwe, and Wang. In particular, our results improve Struwe's H\"older regularity theorem to any H\"older exponent $\alpha\in (0,1)$ when $f\equiv 0$, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of the techniques, we also extend the $L^p$-regularity theory of harmonic maps by Moser to Rivi\`ere-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp.

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