Searcharxiv⌕ Search

arXiv subjects

Youde Wang

Publications and source records attributed to Youde Wang.

At least 37 records · Page 2Linked to original sources
arXiv2023

Gradient estimates for $Δ_pu-|\nabla u|^q+b(x)|u|^{r-1}u=0$ on a complete Riemannian manifold and Liouville type theorems

In this paper the Nash-Moser iteration method is used to study the gradient estimates of solutions to the quasilinear elliptic equation $Δ_p u-|\nabla u|^q+b(x)|u|^{r-1}u=0$ defined on a complete Riemannian manifold $(M,g)$. When $b(x)\equiv0$, a unified Cheng-Yau type estimate of the solutions to this equation is derived. Regardless of whether this equation is defined on a manifold or a region of Euclidean space, certain technical and geometric conditions posed in \cite[Theorem E, F]{MR3261111} are weakened and hence some of the estimates due to Bidaut-Véron, Garcia-Huidobro and Véron (see \cite[Theorem E, F]{MR3261111}) are improved. In addition, we extend their results to the case $p>n=\dim(M)$. When $b(x)$ does not vanish, we can also extend some estimates for positive solutions to the above equation defined on a region of the Euclidean space due to Filippucci-Sun-Zheng \cite{filippucci2022priori} to arbitrary solutions to this equation on a complete Riemannian manifold. Even in the case of Euclidean space, the estimates for positive solutions in \cite{filippucci2022priori} and our results can not cover each other.

math.AP↗
arXiv2023

Rigidity and ε-regularity theorems of Ricci shrinkers

In this paper, we study the rigidity and ε-regularity theorems of Ricci shrinkers. First we prove the rigidity of the asymptotic volume ratio and local volume around a base point of a non-compact Ricci shrinker. Next we obtain some ε-regularity theorems of local entropy and curvature, which improve the previous corresponding results essentially and use them to study the structure of Ricci shrinkers at infinity. Especially, if the curvature of a non-compact Ricci shrinker satisfies some natural integral conditions, then it is asymptotic to a cone.

math.DG↗
arXiv2022

Gradient Estimates For $Δu + a(x)u\log u + b(x)u = 0$ and its Parabolic Counterpart Under Integral Ricci Curvature Bounds

In this paper, we consider a class of important nonlinear elliptic equations $$Δu + a(x)u\log u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$ and $b(x)$ are two $C^1$-smooth real functions. Some new local gradient estimates for positive solutions to these equations are derived by Moser's iteration provided that the integral Ricci curvature is small enough. Especially, some classical results are extended by our estimates and a few interesting corollaries are given. Furthermore, some global gradient estimates are also established under certain geometric conditions. Some estimates obtained in this paper play an important role in a recent paper by Y. Ma and B. Wang [17], which extended and improved the main results due to B. Wang [29] to the case of integral Ricci curvature bounds.

math.DG↗
arXiv2021

Existence and uniqueness of local regular solution to the Schrödinger flow from a bounded domain in $\mathbb{R}^3$ into $\mathbb{S}^2$

In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of Schrödinger flow from a smooth bounded domain $Ω\subset\mathbb{R}^3$ into $\mathbb{S}^2$(namely Landau-Lifshitz equation without dissipation). The proof is built on a parabolic perturbation method, an intrinsic geometric energy argument and some observations on the behaviors of some geometric quantities on the boundary of the domain manifold. It is based on methods from Ding and Wang (one of the authors of this paper) for the Schrödinger flows of maps from a closed Riemannian manifold into a Kähler manifold as well as on methods by Carbou and Jizzini for solutions of the Landau-Lifshitz equation.

math.AP↗
arXiv2021

Very regular solution to Landau-Lifshitz system with spin-polarized transport

In this paper, we provide a precise description of the compatibility conditions for the initial data so that one can show the existence and uniqueness of regular short-time solution to the Neumann initial-boundary problem of a class of Landau-Lifshitz-Gilbert system with spin-polarized transport, which is a strong nonlinear coupled parabolic system with non-local energy.

math.AP↗
arXiv2021

Gradient Estimates And Liouville Theorems For A Class of Nonlinear Elliptic Equations

In this paper, first we study carefully the positive solutions to $Δu+λ_{1}u\ln u +λ_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as Lichnerowicz-type equations, and obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on $(M, g)$. Then, we extend our techniques to a class of more general semilinear elliptic equations $Δu(x)+uh(\ln u)=0$ and obtain some similar results under some suitable analysis conditions on these equations. Moreover, we also obtain some Liouville-type theorems for these equations when $Ric(g)\geq 0$ and establish some Harnack inequalities as consequences.

math.AP↗
arXiv2020

Gradient Estimates For $Δu + au^{p+1}=0$ And Liouville Theorems

In this short note, we use a unified method to consider the gradient estimates of the positive solution to the following nonlinear elliptic equation $Δu + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p <\frac{4}{n}$ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, this improves considerably the previous known results except for the cases $\dim(M)=4$ and supplements the results for the case $\dim(M)\leq 2$. For the case $a<0$ and $p>0$, we also improve considerably the previous related results. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.

math.DG↗
arXiv2020

Yau Type Gradient Estimates For $Δu + au(\log u)^{p}+bu=0$ On Riemannian Manifolds

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M, g)$ $$Δu + au(\log u)^{p}+bu=0,$$ where $a, b\in \mathbb{R}$ and $p$ is a rational number with $p=\frac{k_1}{2k_2+1}\geq2$ where $k_1$ and $k_2$ are positive integer numbers. we obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on $(M, g)$. Our results can be viewed as a natural extension of Yau's estimates on positive harmonic function.

math.DG↗
arXiv2020

Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional

We follow the idea of Wang \cite{W} to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a $n$-dimensional Euclidean domain $\Om$ or a $n$-dimensional closed Riemannian manifold $M$ into a 2-dimensional unit sphere $\U^{2}$. Our conclusions extend a series of related results obtained in the previous literature.

math.AP↗
arXiv2019

Higher order geometric flow of hypersurfaces in a Riemannian manifold

In this paper, we consider the high order geometric flows of a submanifolds $M$ in a complete Riemannian manifold $N$ with $\dim(N)=\dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on $N$. Precisely, we show that if $m\in\mathbb{N}$ is strictly larger than the integer part of $n/2$ and $φ(t)$ is a immersion for all $t\in[0,T)$ and if $\mathfrak{F}_m(φ_0)$ is bounded by a constant which relies on the injectivity radius $\bar{R}>0$ and sectional curvature $\bar{K}_π(\bar{K}_π\leqslant1)$ of $N$ , then $T$ must be $\infty$.

math.DG↗
arXiv2019

Global Weak Solutions to Landau-Lifshitz Equations into Compact Lie Algebras

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz equations from an $n$-dimensional closed Riemannian manifold $\mathbb{T}$ or a bounded domain in $\mathbb{R}^n$ into a unit sphere $S_\mathfrak{g}(1)$ in $\mathfrak{g}$. In particular, we consider the Hamiltonian system associated with the nonlocal energy--{\it micromagnetic energy} defined on a bounded domain of $\mathbb{R}^3$ and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.

math.DG↗
arXiv2018

Local Nonautonomous Schrödinger Flows on Kähler Manifolds

$\,\,\,\,\,\,$In this paper, we prove that the nonautonomous Schrödinger flow from a compact Riemannian manifold into a Kähler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher regularity.

math.DG↗
arXiv2017

Uniqueness of Schrödinger flow on manifolds

In this paper, we show the uniqueness of Schrödinger flow from a general complete Riemannian manifold to a complete Kähler manifold with bounded geometry. While following the ideas of McGahagan[16], we present a more intrinsic proof by using the distance functions and gauge language.

math.DG↗
arXiv2017

Equivariant Schrödinger maps from two dimensional hyperbolic space

In this article, we consider the equivariant Schrödinger map from $\Bbb H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than $4π$, we show that the local existence of Schrödinger map. Furthermore, if the energy of the data sufficiently small, we prove the solutions are global in time.

math.AP↗
arXiv2015

Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the $α_k$-harmonic map sequences with bounded uniformly $α_k$-energy, denoted by $\{u_{α_k}: α_k>1 \quad \mbox{and} \quad α_k\searrow 1\}$, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold is of a positive lower bound and the indices of the $α_k$-harmonic map sequence with respect to the corresponding $α_k$-energy are bounded, then, we can conclude that, if the blow-up phenomena occurs in the convergence of $\{u_{α_k}\}$ as $α_k\searrow 1$, the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence $u_k:(Σ,h_k)\rightarrow N$, where the conformal class defined by $h_k$ diverges, we also prove some similar results.

math.DG↗
arXiv2014

Generalized Bi-Schrödinger Flows and Vortex Filament on Symmetric Lie Algebras

The theory of the vortex filament in three-dimensional fluid dynamics, consisting mainly of the models up to the third-order approximation, is an attractive subject in both physics and mathematics. Many efforts have been devoted to the extension of the theory to higher-dimensional symmetric Lie algebras. However, such a generalization known in literature is strongly restricted by the integrable method. In this article, we endeavor to establish the third-order models of the vortex filament on symmetric Lie algebras in a purely geometric way by generalized bi-Schrödinger flows. Our generalization overcomes the limitation of integrability and creates successfully the desired models on Hermitian or para-Hermitian symmetric Lie algebras. Combining the result in this article with what have been known in literature for the leading-order and the second-order models, we actually exhibit the basic models and the related theory of the vortex filament on symmetric Lie algebras up to the third-order approximation.

math.DG↗
arXiv2013

New Geometric Flows on Riemannian Manifolds and Applications to Schrödinger-Airy Flows

In this paper, we define a class of new geometric flows on a complete Riemannian manifold. The new flow is related to the generalized (third order) Landau-Lifishitz equation. On the other hand it could be thought of a special case of the Schrödinger-Airy flow when the target manifold is a Kähler manifold with constant holomorphic sectional curvature. We show the local existence of the new flow on a complete Riemannian manifold with some assumptions on Ricci tensor. Moreover, if the target manifolds are Einstein or some certain type of locally symmetric spaces, we obtain the global results.

math.DG↗