arXiv subjects

Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 199 records · Page 11Linked to original sources

arXiv2019

Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= u_b \quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} with $u(x,t): \bar Ω\times [0,T) \to S^2$. Here $Ω$ is a bounded, smooth axially symmetric domain in $\mathbb{R}^3$. We prove that for any circle $Γ\subset Ω$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $Γ$, in fact $$ |\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8πδ_Γ\text{ as } t\to T . $$ for a regular function $u_*(x)$, where $δ_Γ$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.

math.AP↗

arXiv2019

Bound state solutions for the supercritical fractional Schrödinger equation

We prove the existence of positive solutions for the supercritical nonlinear fractional Schrödinger equation $(-Δ)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$. We show that if $V(x)=o(|x|^{-2s})$ as $|x|\to +\infty$, then for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions. More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of $(-Δ)^s w=w^p$ in $\mathbb R^n$, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre, Fontelos, González and Wei on the singular fractional Yamabe problem.

math.AP↗

arXiv2019

Axially symmetric solutions of Allen-Cahn equation with finite Morse index

In this paper we study axially symmetric solutions of Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between $4$ and $10$. In dimension $3$, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals $1$.

math.AP↗

arXiv2019

Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume

In this paper we study the following conformally invariant poly-harmonic equation $$Δ^mu=-u^\frac{3+2m}{3-2m}\quad\text{in }\mathbb{R}^3,\quad u>0,$$ with $m=2,3$. We prove the existence of positive smooth radial solutions with prescribed volume $\int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx$. We show that the set of all possible values of the volume is a bounded interval $(0,Λ^*]$ for $m=2$, and it is $(0,\infty)$ for $m=3$. This is in sharp contrast to $m=1$ case in which the volume $\int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx$ is a fixed value.

math.AP↗

arXiv2018

Sign-changing blowing-up solutions for the critical nonlinear heat equation

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $Ω$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in Ω$ and $n\geq 5$. We prove that there exists an integer $k_0$ such that for any integer $k\geq k_0$ there exist initial data $u_0$ and smooth parameter functions $ξ(t)\to q$, $0<μ(t)\to 0$ as $t\to +\infty$ such that the solution $u_q$ of the critical nonlinear heat equation \begin{equation*} \begin{cases} u_t = Δu + |u|^{\frac{4}{n-2}}u\text{ in } Ω\times (0, \infty),\\ u = 0\text{ on } \partial Ω\times (0, \infty),\\ u(\cdot, 0) = u_0 \text{ in }Ω, \end{cases} \end{equation*} has the form \begin{equation*} u_q(x, t) \approx μ(t)^{-\frac{n-2}{2}}\left(Q_k\left(\frac{x-ξ(t)}{μ(t)}\right) - H(x, q)\right), \end{equation*} where the profile $Q_k$ is the non-radial sign-changing solution of the Yamabe equation \begin{equation*} ΔQ + |Q|^{\frac{4}{n-2}}Q = 0\text{ in }\mathbb{R}^n, \end{equation*} constructed in \cite{delpinomussofrankpistoiajde2011}. In dimension 5 and 6, we also prove the stability of $u_q(x, t)$.

math.AP↗

arXiv2018

Infinite time blow-up for the 3-dimensional energy critical heat equation

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension three $$ u_t = Δu + u^5 , \quad {\mbox {in}} \quad \R^3 \times (0,\infty), \ \ u(x, 0)= u_0 (x)\inn \R^3. $$ For each $γ>1$ we find initial data (not necessarily radially symmetric) with $\lim\limits_{r \to \infty} |x|^γu_0 (x) >0$ such that as $t \to \infty$ $$ \| u(\cdot ,t ) \|_\infty \sim t^{γ-1 \over 2} , \quad {\mbox {if}} \quad 1<γ<2, \quad \| u(\cdot ,t ) \|_\infty \sim \sqrt{t}, \quad {\mbox {if}} \quad γ>2, \quad $$ and $$ \| u(\cdot , t)\|_\infty \sim \sqrt{t}\, (\ln t )^{-1} , \quad {\mbox {if}} \quad γ= 2. $$ Furthermore we show that this infinite time blow-up is co-dimensional one stable. The existence of such solutions was conjectured by Fila and King.

math.AP↗

arXiv2018

The Linear Stability of Symmetric Spike Patterns for a Bulk-Membrane Coupled Gierer-Meinhardt Model

We analyze a coupled bulk-membrane PDE model in which a scalar linear 2-D bulk diffusion process is coupled through a linear Robin boundary condition to a two-component 1-D reaction-diffusion (RD) system with Gierer-Meinhardt (nonlinear) reaction kinetics defined on the domain boundary. For this coupled model, in the singularly perturbed limit of a long-range inhibition and short-range activation for the membrane-bound species, asymptotic methods are used to analyze the existence of localized steady-state multi-spike membrane-bound patterns, and to derive a nonlocal eigenvalue problem (NLEP) characterizing $\mathcal{O}(1)$ time-scale instabilities of these patterns. A central, and novel, feature of this NLEP is that it involves a membrane Green's function that is coupled nonlocally to a bulk Green's function. When the domain is a disk, or in the well-mixed shadow-system limit corresponding to an infinite bulk diffusivity, this Green's function problem is analytically tractable, and as a result we will use a hybrid analytical-numerical approach to determine unstable spectra of this NLEP. This analysis characterizes how the 2-D bulk diffusion process and the bulk-membrane coupling modifies the well-known linear stability properties of steady-state spike patterns for the 1-D Gierer-Meinhardt model in the absence of coupling. In particular, phase diagrams in parameter space for our coupled model characterizing either oscillatory instabilities due to Hopf bifurcations, or competition instabilities due to zero-eigenvalue crossings are constructed. Finally, linear stability predictions from the NLEP analysis are confirmed with full numerical finite-element simulations of the coupled PDE system.

nlin.PS↗

arXiv2018

Second order estimates on transition layers

In this paper we establish a uniform $C^{2,θ}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $ n\leq 10$ (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the $C^{2,θ}$ estimate for these level sets to a corresponding one on solutions of Toda system; the other one uses a small regularity theorem on stable solutions of Toda system to establish various decay estimates on these solutions, which gives a lower bound on distances between different sheets of solutions to Toda system or level sets of solutions to Allen-Cahn equation.

math.AP↗

arXiv2018

Type II blow-up in the 5-dimensional energy critical heat equation

We consider the Cauchy problem for the energy critical heat equation $$ u_t = Δu + |u|^{\frac 4{n-2}}u {{\quad\hbox{in } }} \ {\mathbb R}^n \times (0, T), \quad u(\cdot,0) =u_0 {{\quad\hbox{in } }} {\mathbb R}^n $$ in dimension $n=5$. More precisely we find that for given points $q_1, q_2,\ldots, q_k$ and any sufficiently small $T>0$ there is an initial condition $u_0$ such that the solution $u(x,t)$ of the problem blows-up at exactly those $k$ points with rates type II, namely with absolute size $ \sim (T-t)^{-α} $ for $α> \frac 34 $. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

math.AP↗

arXiv2018

A gluing approach for the fractional Yamabe problem with isolated singularities

We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.

math.AP↗

arXiv2018

A compactness theorem of the fractional Yamabe problem, Part I: The non-umbilic conformal infinity

Assume that $(X, g^+)$ is an asymptotically hyperbolic manifold, $(M, [\bar{h}])$ is its conformal infinity, $ρ$ is the geodesic boundary defining function associated to $\bar{h}$ and $\bar{g} = ρ^2 g^+$. For any $γ\in (0,1)$, we prove that the solution set of the $γ$-Yamabe problem on $M$ is compact in $C^2(M)$ provided that convergence of the scalar curvature $R[g^+]$ of $(X, g^+)$ to $-n(n+1)$ is sufficiently fast as $ρ$ tends to 0 and the second fundamental form on $M$ never vanishes. Since most of the arguments in blow-up analysis performed here is irrelevant to the geometric assumption imposed on $X$, our proof also provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem.

math.AP↗

arXiv2018

A complete classification of finite Morse index solutions to elliptic sine-Gordon equation in the plane

The elliptic sine-Gordon equation in the plane has a family of explicit multiple-end solutions (soliton-like solutions). We show that all the finite Morse index solutions belong to this family. We also prove they are non-degenerate in the sense that the corresponding linearized operators have no nontrivial bounded kernel. We then show that solutions with $2n$ ends have Morse index $n(n-1)/2.$

math.AP↗

arXiv2018

Nondegeneracy of harmonic maps from $\mathbb R^2$ to $\mathbb S^2$

We prove that all harmonic maps from $\mathbb R^2$ to $\mathbb S^2$ with finite energy are nondegenerate. That is, for any harmonic map $u$ from $\mathbb R^2$ to $\mathbb S^2$ of degree $m$ (in $\mathbb Z$), all bounded kernel maps of the linearized operator $L_u$ at $u$ are generated by these harmonic maps near $u$ and hence the real dimension of bounded kernel space of $L_u$ is $4|m|+2$.

math.AP↗

arXiv2018

Multi-vortex traveling waves for the Gross-Pitaevskii equation and the Adler-Moser polynomials

For $N\leq34,$ we construct traveling waves with small speed for the Gross-Pitaevskii equation, by gluing $N(N+1)/2$ pairs of degree $\pm1$ vortices of the Ginzburg-Landau equation. The location of these vortices is symmetric in the plane and determined by the Adler-Moser polynomials, which has its origin in the study of Calogero-Moser system and rational solutions of the KdV equation. The construction still works for $N>34$, under the additional assumption that the corresponding Adler-Moser polynomial has no repeated root. It is expected that this assumption holds for any $N\in\mathbb{N}$.

math.AP↗

arXiv2018

Superfluids Passing an Obstacle and Vortex Nucleation

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \[ε^2 Δu+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash Ω, \ \ \frac{\partial u}{\partial ν}=0 \ \mbox{on}\ \partial Ω\] where $ Ω$ is a smooth bounded domain in $ {\mathbb R}^d$ ($d\geq 2$), which is referred as the obstacle and $ ε>0$ is sufficiently small. We first construct a vortex free solution of the form $ u= ρ_ε(x) e^{i \frac{Φ_ε}ε}$ with $ ρ_ε(x) \to 1-|\nabla Φ^δ(x)|^2, Φ_ε(x) \to Φ^δ(x) $ where $Φ^δ(x)$ is the unique solution for the subsonic irrotational flow equation \[ \nabla ( (1-|\nabla Φ|^2)\nabla Φ)=0 \ \mbox{in} \ {\mathbb R}^d \backslash Ω, \ \frac{\partial Φ}{\partial ν} =0 \ \mbox{on} \ \partial Ω, \ \nabla Φ(x) \to δ\vec{e}_d \ \mbox{as} \ |x| \to +\infty \] and $|δ| <δ_{*}$ (the sound speed). In dimension $d=2$, on the background of this vortex free solution we also construct solutions with single vortex close to the maximum or minimum points of the function $|\nabla Φ^δ(x)|^2$ (which are on the boundary of the obstacle). The latter verifies the vortex nucleation phenomena (for the steady states) in superfluids described by the Gross-Pitaevskii equations. Moreover, after some proper scalings, the limits of these vortex solutions are traveling wave solution of the Gross-Pitaevskii equation. These results also show rigorously the conclusions drawn from the numerical computations in \cite{huepe1, huepe2}. Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see \cite{ADP} and references therein) for the trapped Bose-Einstein condensates, are also discussed.

math.AP↗

arXiv2018

Ground States of Two-Component Attractive Bose-Einstein Condensates II: Semi-trivial Limit Behavior

As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-β$ are both attractive, $i.e,$ $a$, $b$ and $β$ are all positive. If $0<b<a^*:=\|w\|^2_2$ and $0<β<a^*$ are fixed, where $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$, the semi-trivial behavior of $(u_1,u_2)$ as $a\nearrow a^*$ is proved in the sense that $u_1$ concentrates at a unique point and while $u_2\equiv 0$ in $R^2$. However, if $0<b<a^*$ and $a^*\leβ<β^*=a^*+\sqrt{(a^*-a)(a^*-b)}$, the refined spike profile and the uniqueness of $(u_1,u_2)$ as $a\nearrow a^*$ are analyzed, where $(u_1,u_2)$ must be unique, $u_1$ concentrates at a unique point, and meanwhile $u_2$ can either blow up or vanish, depending on how $β$ approaches to $a^*$.

math.AP↗

arXiv2018

Gluing methods for vortex dynamics in Euler flows

A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. is that of finding regular solutions with highly concentrated vorticities around $N$ moving {\em vortices}. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. In this paper we devise a {\em gluing approach} for the construction of smooth $N$-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville's equation plus small, more regular terms. Gluing methods have been a powerful tool in geometric constructions by {\em desingularization}. We succeed in applying those ideas in this highly challenging setting.

math.AP↗

arXiv2018

Nondegeneracy, Morse Index and Orbital Stability of the Lump Solution to the KP-I Equation

Let $Q(x,y)= 4 \frac{y^2-x^2+3}{ (x^2+y^2+3)^2}$ be the lump solution of the KP-I equation $$ \partial_x^2 (\partial_x^2 u-u + 3 u^2)-\partial_y^2 u=0.$$ We show that this solution is rigid in the following senses: the only decaying solutions to the linearized operator $$ \partial_x^2 (\partial_x^2 ϕ-ϕ+ 6 Q ϕ)-\partial_y^2 ϕ=0$$ consist of the linear combinations of $ \partial_x Q$ and $ \partial_y Q$. Furthermore we show that the Morse index is exactly one and that it is orbital stable.

math.AP↗