Searcharxiv⌕ Search

arXiv subjects

Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 145 records · Page 8Linked to original sources
arXiv2022

Estimates of bubbling solutions of $SU(3)$ Toda systems at critical parameters-Part 2

In this article we study bubbling solutions of regular $SU(3)$ Toda systems defined on a Riemann surface. There are two major difficulties corresponding to the profile of bubbling solutions: partial blowup phenomenon and bubble accumulation. We prove that when both parameters tend to critical positions, if there is one fully bubbling blowup point, then under one curvature assumption, all the blowup solutions near a blowup point satisfy a spherical Harnack inequality, which completely rules out the bubble-accumulation phenomenon. This fact is crucial for a number of applications.

math.AP↗
arXiv2022

On minima of difference of theta functions and application to hexagonal crystallization

Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ θ(α;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-α\frac{π}{y }|mz+n|^2}$ be the theta function associated with the lattice $L ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following minimization problem of difference of two theta functions \begin{equation}\aligned\nonumber \min_{ \mathbb{H} } \Big(θ(α; z)-βθ(2α; z)\Big) \endaligned\end{equation} where $α\geq 1$ and $ β\in (-\infty, +\infty)$. We prove that there is a critical value $β_c=\sqrt2$ (independent of $α$) such that if $β\leqβ_c$, the minimizer is $\frac{1}{2}+i\frac{\sqrt3}{2}$ (up to translation and rotation) which corresponds to the hexagonal lattice, and if $β>β_c$, the minimizer does not exist. Our result partially answers some questions raised in \cite{Bet2016, Bet2018, Bet2020, Bet2019AMP} and gives a new proof in the crystallization of hexagonal lattice under Yukawa potential.

math.CA↗
arXiv2022

Vanishing Estimates for Liouville equation with quantized singularities

In this article we continue with the research initiated in our previous work on singular Liouville equations with quantized singularity. The main goal of this article is to prove that as long as the bubbling solutions violate the spherical Harnack inequality near a singular source, the first derivatives of coefficient functions must tend to zero.

math.AP↗
arXiv2022

Helmholtz Solutions for the Fractional Laplacian and Other Related Operators

We show that the bounded solutions to the fractional Helmholtz equation, $(-Δ)^s u= u$ for $0<s<1$ in $\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-Δ)u= u$ in $\mathbb{R}^n$ for $n \ge 2$ when $u$ is additionally assumed to be vanishing at $\infty$. When $n=1$, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions $A\cos{x} + B\sin{x}$. We show that this classification of fractional Helmholtz solutions extends for $1<s \le 2$ and $s\in \mathbb{N}$ when $u \in C^\infty(\mathbb{R}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $ψ(-Δ) u= ψ(1)u$ in $\mathbb{R}^n$, when $ψ$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.

math.AP↗
arXiv2021

From KP-I lump solution to travelling waves of Gross-Pitaevskii equation

Let $q(x,y)$ be an nondegenerate lump solution to KP-I (Kadomtsev-Petviashvili-I) equation $$\partial_x^4q-2\sqrt{2}\partial_x^2q-3\sqrt{2}\partial_x((\partial_xq) ^2)-2\partial_y^2q=0. $$ We prove the existence of a traveling wave solution $ u_{\e} (x-ct, y)$ to GP (Gross-Pitaevskii) equation $$ i\partial_{t}Ψ+ΔΨ+(1-|Ψ|^{2})Ψ=0,\ \ \ \mbox{in} \ {\mathbb R}^2 $$ in the transonic limit $$ c=\sqrt{2}-ε^2 $$ with $$ u_ε=1 + i εq(x,y) + {\mathcal O} (ε^2). $$ This proves the existence of finite energy solutions in the so-called Jones-Roberts program in the transonic range $ c \in (\sqrt{2}-ε^2, \sqrt{2})$. The main ingredients in our proof are detailed point-wise estimates of the Green function associated to a family of fourth order hypoelliptic operators $$\partial_x^4-(2\sqrt{2}-\e^2)\partial_x^2-2\partial_y^2+\e^2\partial_x^2\partial_y^2+\e^4\partial_y^4.$$

math.AP↗
arXiv2021

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely monotone functions among lattices is located exactly on a special curve which is part of the boundary of the fundamental region. We also establish a universal result for square lattice being the optimal in certain interval, which is surprising. Our result establishes the hexagonal-rhombic-square-rectangular transition lattice shapes in many physical and biological system (such as Bose-Einstein condensates and two-component Ginzburg-Landau systems). It turns out, our results also apply to locating the minimizers of sum of two Eisenstein series, which is new in number theory.

math.CA↗
arXiv2021

Normalized solutions for Schrödinger system with quadratic and cubic interactions

In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schrödinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on the product of $L^2$-spheres, normalized ground states exist and are obtained as global minimizers. When $N=2$, the energy functional is not always bounded on the product of $L^2$-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on $b_1$ and $b_2$, we prove the existence of normalized solutions. When $N=3$, the energy functional is always unbounded on the product of $L^2$-spheres. We show that under suitable conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as $β\rightarrow 0$. Finally, we deal with the high dimensional cases $N\geq 4$. Several non-existence results are obtained if $β<0$. When $N=4$, $β>0$, the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case $β=0$, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schrödinger system but also leads to a stabilization of the related evolution system.

math.AP↗
arXiv2021

Entire solutions of the magnetic Ginzburg-Landau equation in $\mathbb{R}^4$

We construct entire solutions of the magnetic Ginzburg-Landau equations in dimension 4 using Lyapunov-Schmidt reduction. The zero set of these solutions are close to the minimal submanifolds studied by Arezzo-Pacard\cite{Arezzo}. We also show the existence of a saddle type solution to the equations, whose zero set consists of two vertical planes in $\mathbb{R}^4$. These two types of solutions are believed to be energy minimizers of the corresponding energy functional and lie in the same connect component of the moduli space of entire solutions.

math.AP↗
arXiv2021

Non-degeneracy and quantitative stability of half-harmonic maps from ${\mathbb R}$ to ${\mathbb S}$

We consider half-harmonic maps from $\mathbb{R}$ (or $\mathbb{S}$) to $\mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is $\pm 1$, we prove that the deviation of any map $\boldsymbol{u}:\mathbb{R}\to \mathbb{S}$ from Möbius transformations can be controlled uniformly by $\|\boldsymbol{u}\|_{\dot H^{1/2}(\mathbb{R})}^2-deg \boldsymbol{u}$. This result resembles the quantitative rigidity estimate of degree $\pm 1$ harmonic maps $\mathbb{R}^2\to \mathbb{S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $\boldsymbol{u}$ is already near to a Blaschke product, then the deviation of $\boldsymbol{u}$ to Blaschke products can be controlled by $\|\boldsymbol{u}\|_{\dot H^{1/2}(\mathbb{R})}^2-deg \boldsymbol{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $\boldsymbol{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${\mathbb R}^2\to {\mathbb S}^2$.

math.AP↗
arXiv2021

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.

math.AP↗
arXiv2021

Stability of Caffarelli-Kohn-Nirenberg inequality

In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: \begin{eqnarray*} \bigg(\int_{{\mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq C_{a,b,N}\int_{{\mathbb R}^N}|x|^{-2a}|\nabla u|^2dx \end{eqnarray*} where $N\geq3$, $a<\frac{N-2}{2}$, $a\leq b\leq a+1$ and $p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations $τ^{\frac{N-2}{2}-a}u(τx)$ and scalar multiplications $Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is positive and radially symmetric in the parameter region $b_{FS}(a)\leq b 0$, where $b_{FS}(a)$ is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: \begin{enumerate} \item[$(1)$] \quad stability of CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u, \mathcal{Z})\lesssim\|u\|^2_{D^{1,2}_a({\mathbb R}^N)}-C_{a,b,N}^{-1}\|u\|^2_{L^{p+1}(|x|^{-b(p+1)},{\mathbb R}^N)}$$ where $\mathcal{Z}= \{ c W_τ\mid c\in\bbr\backslash\{0\}, τ>0\}$; \item[$(2)$]\quad stability of CKN inequality in the critical point setting (in the class of nonnegative functions) \begin{eqnarray*} dist_{D_a^{1,2}}(u, \mathcal{Z}_0^ν)\lesssim\left\{\aligned &Γ(u),\quad p>2\text{ or }ν=1,\\ &Γ(u)|\logΓ(u)|^{\frac12},\quad p=2\text{ and }ν\geq2,\\ &Γ(u)^{\frac{p}{2}},\quad 1 0\}.$$

math.AP↗
arXiv2021

Sharp quantitative estimates of Struwe's Decomposition

Suppose $u\in \dot{H}^1(\mathbb{R}^n)$. In a seminal work, Struwe proved that if $u\geq 0$ and $\|Δu+u^{\frac{n+2}{n-2}}\|_{H^{-1}}:=Γ(u)\to 0$ then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwe's decomposition with one bubble in all dimensions, namely $δ(u) \leq C Γ(u)$. For Struwe's decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $δ(u)\leq C Γ(u)$ when $3\leq n\leq 5$ while this is false for $ n\geq 6$. In this paper, we show that \[dist (u,\mathcal{T})\leq C\begin{cases} Γ(u)\left|\log Γ(u)\right|^{\frac{1}{2}}\quad&\text{if }n=6, |Γ(u)|^{\frac{n+2}{2(n-2)}}\quad&\text{if }n\geq 7.\end{cases}\] Furthermore, we show that this inequality is sharp.

math.AP↗
arXiv2021

Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-Δu=λu+μ|u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in H^1(\bbr^N),\quad\int_{\bbr^N}u^2=a^2, \endaligned\right. \end{eqnarray*} where $N\geq3$, $μ>0$, $λ\in\mathbb{R}$ and $2 0$ large. \item[$(3)$]\quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $μ\to0$ and $μ$ goes to its upper bound. \end{enumerate} Our studies answer some questions proposed by Soave in \cite[Remarks~1.1, 1.2 and 8.1]{S20}.

math.AP↗
arXiv2021

Multi-Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain

In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions. In particular we rigorously prove the existence of symmetric and asymmetric two-spike solutions using a Lyapunov-Schmidt reduction. The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain $2\times 2$ matrix. Our rigorous results are complemented by formal calculations of $N$-spike solutions using the method of matched asymptotic expansions. In addition, we explicitly consider examples of one- and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds. By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities. Furthermore, when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity. However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.

nlin.PS↗
arXiv2021

Generalized Adler-Moser Polynomials and Multiple vortex rings for the Gross-Pitaevskii equation

New finite energy traveling wave solutions with small speed are constructed for the three dimensional Gross-Pitaevskii equation \begin{equation*} iΨ_t= ΔΨ+(1-|Ψ|^2)Ψ, \end{equation*} where $Ψ$ is a complex valued function defined on ${\mathbb R}^3\times{\mathbb R}$. These solutions have the shape of $2n+1$ vortex rings, far away from each other. Among these vortex rings, $n+1$ of them have positive orientation and the other $n$ of them have negative orientation. The location of these rings are described by the roots of a sequence of polynomials with rational coefficients. The polynomials found here can be regarded as a generalization of the classical Adler-Moser polynomials and can be expressed as the Wronskian of certain very special functions. The techniques used in the derivation of these polynomials should have independent interest.

math.AP↗
arXiv2021

Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation

We consider the energy critical semilinear heat equation $$ \left\{\begin{aligned} &\partial_t u-Δu =|u|^{\frac{4}{n-2}}u &\mbox{in } {\mathbb R}^n\times(0,T),\\ &u(x,0)=u_0(x), \end{aligned}\right. $$ where $ n\geq 3$, $u_0\in L^\infty({\mathbb R}^n)$, and $T\in {\mathbb R}^+$ is the first blow up time. We prove that if $ n \geq 7$ and $ u_0 \geq 0$, then any blowup must be of Type I, i.e., \[\|u(\cdot, t)\|_{L^\infty({\mathbb R}^n)}\leq C(T-t)^{-\frac{1}{p-1}}.\] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysis of bubbling behavior (bubbling tower/cluster).

math.AP↗