arXiv subjects

Zaihui Gan

Publications and source records attributed to Zaihui Gan.

7 recordsLinked to original sources

arXiv2026

$L^{2}$-Supercritical Nonlinear Klein-Gordon System with Quadratic Asymmetric Interaction

In this paper we investigate the global existence, blow-up and standing waves for the $L^{2}$-supercritical nonlinear Klein-Gordon equations with quadratic asymmetric interaction (LSNKG). First, by introducing a suitable auxiliary functional, using concavity analysis and virial estimates, we obtain a finite time blow-up result for solutions to the Cauchy problem of (LSNKG) when the initial energy is negative. Next, by defining appropriate functionals, manifolds and a constrained variational problem, we employ variational method and the Lagrange multiplier method to derive the existence of ground state solutions for the corresponding nonlinear elliptic (steady-state) system, thereby to establish the existence of standing wave with the ground state for (LSNKG). Then, using the variational characterization of the ground state solutions and constructing invariant sets under the flow generated by the Cauchy problem for (LSNKG), we combine the potential well argument with concavity analysis to establish a sharp threshold between blow-up in finite time and global existence. Finally, by exploiting the variational characterization of the ground state, introducing appropriate scalings, and choosing suitable initial data based on the ground state, we justify the instability of standing wave with the ground state for (LSNKG).

math.AP↗

arXiv2025

Improved Berezin-Li-Yau inequality and Kröger inequality and consequences

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kröger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kröger's inequality resolves an open question raised by Weidl from 2006. The improvements allow us to show that, for any open bounded domains, there are infinite many Dirichlet eigenvalues satisfying Pólya's conjecture if $n\ge 3$, and infinite many Neumann eigenvalues satisfying Pólya's conjecture if $n\ge 5$ and the Neumann spectrum is discrete.

math.SP↗

arXiv2023

Existence and Instability of Standing Wave for the Two-wave Model with Quadratic Interaction

In this paper, we establish the existence and instability of standing wave for a system of nonlinear Schrödinger equations arising in the two-wave model with quadratic interaction in higher space dimensions under mass resonance conditions. Here, we eliminate the limitation for the relationship between complex constants $a_{1}$ and $a_{2}$ given in \cite{HOT}, and consider arbitrary real positive constants $a_{1}$ and $a_{2}$. First of all, according to the conservation identities for mass and energy, using the so-called virial type estimate, we obtain that the solution for the Cauchy problem under consideration blows up in finite time in $H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})$ with space dimension $N\geq 4$. Next, for space dimension $N$ with $4<N<6$, we establish the existence of the ground state solution for the elliptic equations corresponding to the nonlinear Schrödinger equations under the frequency and mass resonance by adopting variational method, and further achieve the exponential decay at infinity for the ground state. This implies the existence of standing wave for the nonlinear Schrödinger equaitons under consideration. Finally, by defining another constrained minimizing problems for a pair of complex-valued functions, a suitable manifold, referring to the characterization of the standing wave, making appropriate scaling and adopting virial type estimate, we attain the instability of the standing wave for the equations under frequency and mass resonance in space dimension $N$ with $4<N<6$ by virtue of the conservations of mass and energy. Here, we adopt the equivalence of two constrained minimizing problems defined for pairs of complex-valued and real-valued functions $(u,v)$, respectively, when $(u,v)$ is a pair of real-valued functions.

math.AP↗

arXiv2023

Sharp Lower Bound for the Blow-up Rate of Solutions to the Magnetic Zakharov System without the Skin Effect

In this paper, we consider the Cauchy problem of the magnetic Zakharov system in two-dimensional space: \[ \begin{cases} & i E_{1t}+ΔE_1-n E_1+ηE_2 (E_1\overline{E_2}-\overline{E_1} E_2)=0, \\ & i E_{2t}+ΔE_2-n E_2+ηE_1(\overline{E_1} E_2-E_1\overline{E_2})=0, \\ & n_t+\nabla \cdot \textbf{v}=0, \\ & \textbf{v}_t+\nabla n+\nabla (|E_1|^2+|E_2|^2)=0, \\ \end{cases} \tag{G-Z} \] with initial data $\left(E_{10}(x),E_{20}(x),n_{0}(x),\mathbf{v}_{0}(x)\right)$, which describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, where $η>0$ is a physical constant coefficient. The two nonlinear terms generated by the cold magnetic field bring in a different difficulty from that for the classical Zakharov system. Assuming the initial mass satisfies the following estimates: \begin{gather*} \frac{||Q||_{L^2(\mathbb{R}^2)}^2}{1+η} <||E_{10}||_{L^2(\mathbb{R}^2)}^2+||E_{20}||_{L^2(\mathbb{R}^2)}^2 <\frac{||Q||_{L^2(\mathbb{R}^2)}^2}η, \end{gather*} where $Q$ is the unique radially positive solution of the equation $-ΔV+V=V^3 $, we prove that there is a constant $c>0$ depending only on the initial data such that for $t$ near $T$ (the blow-up time), \begin{gather*} \left\|\left(E_1,E_2,n,\textbf{v}\right)\right\|_{H^1(\mathbb{R}^2)\times H^1(\mathbb{R}^2)\times L^2(\mathbb{R}^2)\times L^2(\mathbb{R}^2)}\geqslant \frac{c}{ T-t }. \end{gather*} As the magnetic coefficient $η$ tends to $0$, the blow-up rate recovers the result for the classical 2-D Zakharov system due to Merle \cite{25Frank}. For any size positive $η$, under the current assumption on the initial mass, we give a mathematically rigorous justification for the fact that the presence of magnetic effects without the skin effect in the cold plasma does not change the optimal lower bound for the blow-up rates.

math.AP↗

arXiv2019

On the Viscous Camassa-Holm Equations with Fractional Diffusion

We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be $2s$ with $s\in [n/4,1)$, which seems to be sharp for the validity of the main results of the paper; here $n=2,3$ is the dimension of space. We prove global well-posedness in $C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2}))$ whenever the initial data $u_0\in D(A)$, where $A$ is the Stokes operator. We also prove that such global solutions gain regularity instantaneously after the initial time. A bound on a higher-order spatial norm is also obtained.

math.AP↗

arXiv2018

Large Time Behavior and Convergence for the Camassa-Holm Equations with Fractional Laplacian Viscosity

In this paper, we consider the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity in the whole space. In stark contrast to the Camassa-Holm equations without any nonlocal effect, to our best knowledge, little has been known on the large time behavior and convergence for the nonlocal equations under study. We first study the large time behavior of solutions. We then discuss the relation between the equations under consideration and the imcompressible Navier-Stokes equations with fractional Laplacian viscosity (INSF). The main difficulty to achieve them lies in the fractional Laplacian viscosity. Fortunately, by employing some properties of fractional Laplacian, in particular, the fractional Leibniz chain rule and the fractional Gagliardo-Nirenberg-Sobolev type estimates, the high and low frequency splitting method and the Fourier splitting method, we first establish the large time behavior concerning non-uniform decay and algebraic decay of solutions to the nonlocal equations under study. In particular, under the critical case $s=\dfrac{n}{4}$, the nonlocal version of Ladyzhenskaya's inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the non-uniform decay and algebraic decay. On the other hand, by means of the fractional heat kernel estimates, we figure out the relation between the nonlocal equations under consideration and the equations (INSF). Specifically, we prove that the solution to the Camassa-Holm equations with nonlocal viscosity converges strongly as the filter parameter $α\rightarrow~0$ to a solution of the equations (INSF).

math.AP↗

arXiv2018

Regularity of Solutions of the Camassa-Holm Equations with Fractional Laplacian Viscosity

We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $\displaystyle s=\frac{n}{4}$ with $n=2,3$, we will make extra efforts for acquiring the expected estimates as obtained in the case $\displaystyle \frac{n}{4}<s<1$. By the aid of the fractional Leibniz rule and the nonlocal version of Ladyzhenskaya's inequality, we prove the existence, uniqueness and regularity to the Camassa-Holm equations under study by the energy method and a bootstrap argument, which rely crucially on the fractional Laplacian viscosity. In particular, under the critical case $s=\dfrac{n}{4}$, the nonlocal version of Ladyzhenskaya's inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the desired results concernig existence, uniqueness and regularity.

math.AP↗