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Zexi Wang

Publications and source records attributed to Zexi Wang.

18 recordsLinked to original sources
arXiv2026

Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices

We construct weak solutions of the three-dimensional incompressible Navier--Stokes equations on the torus. The convex-integration scheme is based on the moving-dipole construction of Bru\`e, Colombo, and Kumar~\cite{BrueColomboKumar2024}. For the explicit exponent $\bar p=\frac65+5\times10^{-5},$ and for any two mean-zero, divergence-free vector fields in $L^2(\mathbb T^3)$, we construct a weak solution whose traces at times $0$ and $1$ approximate the prescribed fields arbitrarily well and which satisfies \[ u\in C([0,1];L^2(\mathbb T^3)), \qquad \nabla u\in C([0,1];L^{\bar p}(\mathbb T^3)). \] Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in \(L^2_\sigma(\mathbb T^3)\).The principal perturbations are localized, rescaled copies of Hill's spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the \(L^{6/5}\)-scale of the velocity gradient. The construction uses localization of the potential exterior flow, long-orbit averaging of moving vortex cores, an auxiliary source correction, and a temporal corrector compatible with uniform-in-time Sobolev control.

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arXiv2026

Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group

We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{nS} S_{HL}(Q,\mu)\left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}} \frac{|u(\xi)|^{Q^{\ast}_{\mu}}|u(\eta)|^{Q^{\ast}_{\mu}}} {|\eta^{-1}\xi|^{\mu}}\,d\xi d\eta\right)^{\frac{1}{Q^{\ast}_{\mu}}} \leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}}u|^{2}d\xi, \qquad u\in S^{1,2}(\mathbb{H}^{n}), \end{equation} where $Q=2n+2$, $n\geq1$, $0<\mu H_{BE}$ with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of $H_{NS}$.

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arXiv2026

Remainder terms and sharp quantitative stability for a nonlocal Sobolev inequality on the Heisenberg group

In this paper, we study the following nonlocal Sobolev inequality on the Heisenberg group \begin{equation}\label{eq:HLS} S_{HL}(Q,\mu) \left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{|u(\xi)|^{Q^{\ast}_{\mu}}|u(\eta)|^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}{d}\xi{d}\eta\right)^{\frac{1}{Q^{\ast}_{\mu}}}\leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}}u|^{2}d\xi,\quad \forall \, u\in S^{1,2}(\mathbb{H}^{n}), \end{equation} where $Q=2n+2$ is the homogeneous dimension of the Heisenberg group $\mathbb{H}^{n}$, $n\geq1$, $\mu\in(0,Q)$, $Q^{\ast}_{\mu}=\frac{2Q-\mu}{Q-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and the Folland-Stein-Sobolev inequality on the Heisenberg group, $S_{HL}(Q,\mu)$ is the sharp constant of \eqref{eq:HLS}, and $S^{1,2}(\mathbb{H}^{n})$ is the Folland-Stein-Sobolev space. %of the nonlocal-Sobolev inequality. It is well-known that, up to a translation and suitable scaling, \begin{equation}\label{eq:abs} -\Delta_{\mathbb{H}} u=\left(\int_{\mathbb{H}^{n}}\frac{|u(\eta)| ^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}{d}\eta\right)|u|^{Q_\mu^*-2}u,~~u\in S^{1,2}(\mathbb{H}^{n}) \end{equation} is the Euler-Lagrange equation corresponding to the associated minimization problem. On the one hand, we show the existence of a gradient-type remainder term for inequality \eqref{eq:HLS} when $Q\geq4$, $\mu\in (0,4]$, and as a corollary, derive the existence of a remainder term in the weak $L^{\frac{Q}{Q-2}}$-norm on bounded domains. On the other hand, we establish the quantitative stability of critical points for equation \eqref{eq:abs} in the multi-bubble case when $Q=4$ and $\mu\in (2,4)$.

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arXiv2025

An extension of Cabr\'{e}-Chanillo theorem to the $p$-laplacian

In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\\ u>0&in\ \Om,\\ u=0&on\ \partial\Om, \end{cases} \end{equation*} where $p>2$, $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Om\subset\R^2$ is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution $u$ admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, $Argmax(u)$ is a point or segment.

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arXiv2024

Construction of solutions for the critical polyharmonic equation with competing potentials

In this paper, we consider the following critical polyharmonic equation \begin{align*}%\label{abs} ( -\Delta)^m u+V(|y'|,y'')u=Q(|y'|,y'')u^{m^*-1},\quad u>0, \quad y=(y',y'')\in \mathbb{R}^3\times \mathbb{R}^{N-3}, \end{align*} where $N>4m+1$, $m\in \mathbb{N}^+$, $m^*=\frac{2N}{N-2m}$, $V(|y'|,y'')$ and $Q(|y'|,y'')$ are bounded nonnegative functions in $\mathbb{R}^+\times \mathbb{R}^{N-3}$. By using the reduction argument and local Poho\u{z}aev identities, we prove that if $Q(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $Q(r_0,y_0'')>0$, $D^\alpha Q(r_0,y_0'')=0$ for any $|\alpha|\leq 2m-1$ and $B_1V(r_0,y_0'')-B_2\sum\limits_{|\alpha|=2m}D^\alpha Q(r_0,y_0'')\int_{\mathbb{R}^N}y^\alpha U_{0,1}^{m^*}dy>0$, then the above problem has a family of solutions concentrated at points lying on the top and the bottom circles of a cylinder, where $B_1$ and $B_2$ are positive constants that will be given later.

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arXiv2024

Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three

In this paper, we are interested in the existence of solutions for the following Choquard type Brezis-Nirenberg problem \begin{align*} \left\{ \begin{array}{ll} -\Delta u=\displaystyle\Big(\int\limits_{\Omega}\frac{u^{6-\alpha}(y)}{|x-y|^\alpha}dy\Big)u^{5-\alpha}+\lambda u, \ \ &\mbox{in}\ \Omega, u=0, \ \ &\mbox{on}\ \partial \Omega, \end{array} \right. \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$, $\alpha\in (0,3)$, $6-\alpha$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and $\lambda$ is a real positive parameter. By applying the reduction argument, we find and characterize a positive value $\lambda_0$ such that if $\lambda-\lambda_0>0$ is small enough, then the above problem admits a solution, which blows up and concentrates at the critical point of the Robin function as $\lambda\rightarrow \lambda_0$. Moreover, we consider the above problem under zero Neumann boundary condition.

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arXiv2024

New type of solutions for the critical polyharmonic equation

In this paper, we consider the following critical polyharmonic equation \begin{align*}%\label{abs} ( -Δ)^m u+V(|y'|,y'')u=u^{m^*-1},\quad u>0, \quad y=(y',y'')\in \mathbb{R}^3\times \mathbb{R}^{N-3}, \end{align*} where $m^*=\frac{2N}{N-2m}$, $N>4m+1$, $m\in \mathbb{N}^+$, and $V(|y'|,y'')$ is a bounded nonnegative function in $\mathbb{R}^+\times \mathbb{R}^{N-3}$. By using the reduction argument and local Pohouzaev identities, we prove that if $r^{2m}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, then the above problem has a new type of solutions, which concentrate at points lying on the top and the bottom circles of a cylinder.

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arXiv2024

New Regularity Criteria for Navier-Stokes and SQG Equations in Critical Spaces

In this paper, we investigate some priori estimates to provide the critical regularity criteria for incompressible Navier-Stokes equations on $\mathbb{R}^3$ and super critical surface quasi-geostrophic equations on $\mathbb{R}^2$. Concerning the Navier-Stokes equation, we demonstrate that a Leray-Hopf solution $u$ is regular if $u\in L_T^{\frac{2}{1-α}} \dot{B}^{-α}_{\infty,\infty}(\mathbb{R}^3)$, or $u$ in Lorentz space $ L_T^{p,r} \dot{B}^{-1+\frac{2}{p}}_{\infty,\infty}(\mathbb{R}^3)$, with $4\leq p\leq r<\infty$. Additionally, an alternative regularity condition is expressed as $u\in L_{T}^{\frac{2}{1-α}} \dot{B}^{-α}_{\infty,\infty}(\mathbb{R}^3)+{L_T^\infty\dot{B}^{-1}_{\infty,\infty}}(\mathbb{R}^3)$($α\in(0,1)$), contingent upon a smallness assumption on the norm $L_T^\infty\dot{B}^{-1}_{\infty,\infty}$. For the SQG equation, we derive that a Leray-Hopf weak solution $θ\in L_T^{\fracα{\varepsilon}} \dot{C}^{1-α+ε}(\mathbb{R}^2)$ is smooth for any $\varepsilon$ small enough. Similar to the case of Navier-Stokes equation, we derive regularity criterion in more refined spaces, i.e. Lorentz spaces $L_T^{\fracαε,r}\dot{C}^{1-α+ε}(\mathbb{R}^2)$ and addition of two critical spaces $L_{T}^{\fracαε}\dot{C}^{1-α+ε}(\mathbb{R}^2)+{L_T^\infty\dot{C}^{1-α}(\mathbb{R}^2)}$, with smallness assumption on $L_T^\infty\dot{C}^{1-α}(\mathbb{R}^2)$.

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arXiv2023

Blow-up solutions concentrated along minimal submanifolds for asymptotically critical Lane-Emden systems on Riemannian manifolds

Let $(\mathcal{M},g)$ and $(\mathcal{K},κ)$ be two Riemannian manifolds of dimensions $N$ and $m$, respectively. Let $ω\in C^2(\mathcal{M})$, $ω>0$. The warped product $\mathcal{M}\times_ω\mathcal{K}$ is the $(N+m)$-dimensional product manifold $\mathcal{M}\times \mathcal{K}$ furnished with metric $g+ω^2κ$. We are concerned with the following elliptic system $$\begin{align}\label{yuanshi} \left\{ \begin{array}{ll} -Δ_{g+ω^2κ} u+h(x)u=v^{p-α\varepsilon}, \ \ &\mbox{in $(\mathcal{M}\times_ω\mathcal{K},g+ω^2κ)$},\\ -Δ_{g+ω^2κ} v+h(x)v=u^{q-β\varepsilon}, \ \ &\mbox{in $(\mathcal{M}\times_ω\mathcal{K},g+ω^2κ)$},\\ u,v>0, \ \ &\mbox{in $(\mathcal{M}\times_ω\mathcal{K},g+ω^2κ)$}, \end{array} \right.\qquad(0.1)\end{align}$$ where $Δ_{g+ω^2κ} = div_{g+ω^2κ} \nabla$ is the Laplace-Beltrami operator on $\mathcal{M}\times_ω\mathcal{K}$, $h(x)$ is a $C^1$-function on $\mathcal{M}\times_ω\mathcal{K}$, $\varepsilon>0$ is a small parameter, $α,β>0$ are real numbers, $\varepsilon$ is a positive parameter, $(p,q)\in (1,+\infty)\times (1,+\infty)$ satisfies $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$. For any given integer $k\geq2$, using the Lyapunov-Schmidt reduction, we prove that problem (0.1) has a $k$-peaks solution concentrated along a $m$-dimensional minimal submanifold of $(\mathcal{M}\times_ω\mathcal{K})^k$.

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arXiv2023

Multiple blowing-up solutions for asymptotically critical Lane-Emden systems on Riemannian manifolds

Let $(\mathcal{M},g)$ be a smooth compact Riemannian manifold of dimension $N\geq 8$. We are concerned with the following elliptic system \begin{align*} \left\{ \begin{array}{ll} -Δ_g u+h(x)u=v^{p-α\varepsilon}, \ \ &\mbox{in}\ \mathcal{M}, -Δ_g v+h(x)v=u^{q-β\varepsilon}, \ \ &\mbox{in}\ \mathcal{M}, u,v>0, \ \ &\mbox{in}\ \mathcal{M}, \end{array} \right. \end{align*} where $Δ_g=div_g \nabla$ is the Laplace-Beltrami operator on $\mathcal{M}$, $h(x)$ is a $C^1$-function on $\mathcal{M}$, $\varepsilon>0$ is a small parameter, $α,β>0$ are real numbers, $(p,q)\in (1,+\infty)\times (1,+\infty)$ satisfies $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$. Using the Lyapunov-Schmidt reduction method, we obtain the existence of multiple blowing-up solutions for the above problem.

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arXiv2023

Normalized solutions for a fractional $N/s$-Laplacian Choquard equation with exponential critical nonlinearities

In this paper, we are concerned with the following fractional $N/s$-Laplacian Choquard equation \begin{align*} \begin{cases} (-Δ)^s_{N/s}u=λ|u|^{\frac{N}{s}-2}u +(I_μ*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}|u|^{N/s} \mathrm{d}x=a^{N/s}, \end{cases} \end{align*} where $s\in(0,1)$, $1<\frac{N}{s}\in \mathbb{N}^+$, $a>0$ is a prescribed constant, $λ\in \mathbb{R}$, $I_μ(x)=\frac{1}{|x|^μ}$ with $μ\in(0,N)$, $F$ is the primitive function of $f$, and $f$ is a continuous function with exponential critical growth of Trudinger-Moser type. Under some suitable assumptions on $f$, we prove that the above problem admits a ground state solution for any given $a>0$, by using the constraint variational method and minimax technique.

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arXiv2023

Normalized ground states for a fractional Choquard system in $\mathbb{R}$

In this paper, we study the following fractional Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} (-Δ)^{1/2}u=λ_1 u+(I_μ*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}, (-Δ)^{1/2}v=λ_2 v+(I_μ*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2\mathrm{d}x=a^2,\quad \displaystyle\int_{\mathbb{R}}|v|^2\mathrm{d}x=b^2,\quad u,v\in H^{1/2}(\mathbb{R}), \end{array} \right. \end{split} \end{align*} where $(-Δ)^{1/2}$ denotes the $1/2$-Laplacian operator, $a,b>0$ are prescribed, $λ_1,λ_2\in \mathbb{R}$, $I_μ(x)=\frac{1}{|x|^μ}$ with $μ\in(0,1)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential critical growth in $\mathbb{R}$. By using a minimax principle and analyzing the monotonicity of the ground state energy with respect to the prescribed masses, we obtain at least one normalized ground state solution for the above system.

math.AP↗
arXiv2023

Normalized solutions for a fractional Choquard-type equation with exponential critical growth in $\mathbb{R}$

In this paper, we study the following fractional Choquard-type equation with prescribed mass \begin{align*} \begin{cases} (-Δ)^{1/2}u=λu +(I_μ*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2 \mathrm{d}x=a^2, \end{cases} \end{align*} where $(-Δ)^{1/2}$ denotes the $1/2$-Laplacian operator, $a>0$, $λ\in \mathbb{R}$, $I_μ(x)=\frac{1}{|x|^μ}$ with $μ\in(0,1)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Trudinger-Moser inequality. By using a minimax principle based on the homotopy stable family, we obtain that there is at least one normalized ground state solution to the above equation.

math.AP↗
arXiv2023

Normalized ground states for a biharmonic Choquard system in $\mathbb{R}^4$

In this paper, we study the existence of normalized ground state solutions for the following biharmonic Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} Δ^2u=λ_1 u+(I_μ*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}^4, Δ^2v=λ_2 v+(I_μ*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad \displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $a,b>0$ are prescribed, $λ_1,λ_2\in \mathbb{R}$, $I_μ=\frac{1}{|x|^μ}$ with $μ\in (0,4)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential subcritical or critical growth in the sense of the Adams inequality. By using a minimax principle and analyzing the behavior of the ground state energy with respect to the prescribed mass, we obtain the existence of ground state solutions for the above problem.

math.AP↗
arXiv2023

Normalized solutions to the biharmonic nonlinear Schrödinger equation with combined nonlinearities

In this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schrödinger equation with combined nonlinearities \begin{equation*} Δ^2u=λu+μ|u|^{q-2}u+|u|^{p-2}u,\quad \text {in $\mathbb{R}^N$} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{equation*} where $N\geq2$, $μ\in \mathbb{R}$, $a>0$, $2 0$, $p=4^*$, and $2<q<2+\frac{8}{N}$. Moreover, we also consider the case $μ=0$ and $μ<0$.

math.AP↗
arXiv2022

Normalized ground states for a biharmonic Choquard equation with exponential critical growth

In this paper, we consider the normalized ground state solution for the following biharmonic Choquard type problem \begin{align*} \begin{split} \left\{ \begin{array}{ll} Δ^2u-βΔu=λu+(I_μ*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $β\geq0$, $c>0$, $λ\in \mathbb{R}$, $I_μ=\frac{1}{|x|^μ}$ with $μ\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Adams inequality. By using a minimax principle based on the homotopy stable family, we obtain that the above problem admits at least one ground state normalized solution.

math.AP↗
arXiv2022

Normalized solutions for a biharmonic Choquard equation with exponential critical growth in $\mathbb{R}^4$

In this paper, we study the following biharmonic Choquard type equation \begin{align*} \begin{split} \left\{ \begin{array}{ll} γΔ^2u-βΔu=λu+(I_μ*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2>0,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $γ>0$, $β\geq0$, $λ\in \mathbb{R}$, $I_μ=\frac{1}{|x|^μ}$ with $μ\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth. We can prove the existence of ground state normalized solutions for the above problem when the nonlinearity $f$ satisfies some conditions.

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