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Zihua Guo

Publications and source records attributed to Zihua Guo.

At least 19 recordsLinked to original sources
arXiv2026

Global dynamics above the ground state energy for the 3D Zakharov system

We study the global dynamics for the 3D Zakharov system with radial initial data of energy slightly above the ground state energy, proving that the initial data set splits into nine nonempty, pairwise disjoint regions in which the solutions have distinct behaviors: growup, scattering, or trapped by the ground state, as time goes in the forward or backward directions. It is a classification similar to those for the nonlinear Klein-Gordon equation and for the nonlinear Schr\"odinger equation, extending the previous results on the Zakharov system below the ground state energy. The proof relies on the normal form technique to handle the quadratic frequency interactions and careful modulational analysis around the ground state. One novelty is a new family of localized virial estimates with monotonicity. These virial estimates are crucial for us to study the dynamics away from the ground state and may be of independent interest for other purposes.

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arXiv2026

Well-posedness and vanishing rotational limit for the rotating incompressible Navier-Stokes equations in hybird Besov space

We establish the well-posedness of the 3D rotating incompressible Navier-Stokes equations with critical initial data $u_{0,\Omega}\in X_{0,q,p}^{\Omega}$ for $p<5$, where $X_{0,q,p}^{\Omega}$ is defined by the norm \begin{equation*} \begin{aligned} &\|u_{0,\Omega}\|_{X_{0,q,p}^{\Omega}}:= \Omega^{3- \frac{6}{q}}\|u_{0,\Omega}\|_{\dot{B}_{q,\infty}^{-7+\frac{15}{q}}}^{\ell_\Omega} +\|u_{0,\Omega}\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}^{h_\Omega}. \end{aligned} \end{equation*} This extends the previous results by Chen, Miao, and Zhang (\cite{CMZ2013}). The main ingredients are a new global-in-time dissipative-dispersive estimate for the Stokes--Coriolis semigroup and corresponding bilinear estimates. Furthermore, we establish the vanishing rotational limit for the 3D rotating Navier-Stokes equations as $\Omega\rightarrow 0^{+}$.

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arXiv2026

Sharp Strichartz estimate for the 1D periodic Schrödinger equation

We prove the following estimate \[ \|{e^{it\partial_x^2}f}\|_{L_{(t,x)\in \mathbb{T}^2}^6}\leq C (\log N)^{1/6} \|f\|_{L^2_x(\mathbb{T})}, \] assuming $\mbox{supp} (\hat f)\subset [-N,N]$ for $N>1$. The bound $(\log N)^{1/6}$ is sharp in view of the lower bound by Bourgain \cite{Bourgain}.

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arXiv2026

On the well-posedness of the KP-I equation

We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence of perturbations with finite energy of non decaying smooth global solutions.

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arXiv2026

On the well-posedness of the compressible Navier-Stokes equations

We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space $\mathbb{X}_p=\dot{B}_{p,1}^{\frac{d}{p}}\times \dot{B}_{p,1}^{-1+\frac{d}{p}}$ for $1\leq p<2d$ with $d\geq2$. The main new result is the continuity of the solution maps from $\mathbb{X}_p$ to $C([0,T]: \mathbb{X}_p)$, which was not proved in previous works \cite{D2001, D2005, D2014}. To prove our results, we derive a new difference estimate in $L_t^1L_x^\infty$. Then we combine the method of frequency envelope (see \cite{Tao04}) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see \cite{D2014}). As a by-product, the Lagrangian transform $(a,u)\to (\bar a, \bar u)=(a\circ X, u\circ X)$ used in \cite{D2014} is a continuous bijection and hence bridges the Eulerian and Lagrangian methods.

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arXiv2025

Global solutions to 3D quadratic nonlinear Schrödinger-type equation

We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us more flexibilities in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.

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arXiv2025

Global well-posedness for the 3D compressible Navier-Stokes equations in optimal Besov space

We consider the Cauchy problem to the 3D barotropic compressible Navier-Stokes equation. We prove global well-posedness, assuming that the initial data $(ρ_0-1,u_0)$ has small norms in the critical Besov space $\mathbb{X}_p=\dot{B}_{p,1}^{3/p}(\mathbb{R}^3)\times \dot{B}_{p,1}^{-1+3/p}(\mathbb{R}^3)$ for $2\leq p<6$ and $(ρ_0-1,ρ_0u_0)$ satisfies an additional low frequency condition. Our results extend the previous results in \cite{FD2010, CMZ2010, H20112} where $p<4$ is needed for high frequency, to the optimal range $p<6$. The main ingredients of the proof consist of: a novel nonlinear transform that uses momentum formulation for low-frequency and effective velocity method for high frequency, and estimate of parabolic-dispersive semigroup that enables a $L^q$-framework for low frequency.

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arXiv2025

Scattering for the Klein-Gordon-Zakharov system in two dimensions

We study the Klein-Gordon-Zakharov system in two spatial dimensions, an important model in plasma physics. For small, smooth, and spatially localized initial data, we establish the global existence of solutions and characterize their sharp long-time behavior, including sharp time decay and scattering properties. A particularly interesting phenomenon is that the Klein-Gordon component exhibits modified scattering for certain initial data, while for others it undergoes linear scattering-a dichotomy highlighting delicate long-range interaction effects. The major obstacles are lack of symmetry and weak decay of the solution in two dimensions. To overcome these, we introduce a novel nonlinear transformation of the wave component and reinterpret the nonlinear coupling as a perturbation of the mass term in the Klein-Gordon equation. The proof employs a combination of physical space and frequency space methods.

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arXiv2025

On the convergence to the Navier-Stokes-Maxwell system with solenoidal Ohm's law

The incompressible Navier-Stokes-Maxwell system with solenoidal Ohm's law can be viewed as as the asymptotic limit of the two-fluid incompressible Navier-Stokes-Maxwell system as the momentum transfer coefficient tends to zero (see [1], Arsénio, Ibrahim and Masmoudi, Arch. Ration. Mech. Anal., 2015). We prove this limit rigorously without loss of regularity by using the idea of frequency envelope.

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arXiv2025

Complex-valued solutions of the mKdV equations in generalized Fourier-Lebesgue spaces

We study the \emph{complex-valued} solutions to the Cauchy problem of the modified Korteweg-de Vries equation on the real line. To study the low-regularity problems, we employ a generalized Fourier-Lebesgue space $\widehat{M}^{s}_{r,q}(\mathbb{R})$ that unifies the modulation spaces and the Fourier-Lebesgue spaces. We then prove sharp local well-posedness results in this space by perturbation arguments using $X^{s,b}$-type spaces. Our results improve the previous one in \cite{GV}.

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arXiv2024

Uniform estimates for oscillatory integrals with parameter-dependent phases

We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some $L^p$-estimate for the remainder term and get various uniform estimates when the phase functions satisfy certain conditions. This enables us to reduce the requirement of the smoothness on the phase functions, and hence improve the results in \cite[Theorem 7.7.5]{Hormander} and also obtain a refined version of the well-known Van der Corput Lemma. Some applications on the uniform expansion of the Bessel functions and dispersive estimates are also given.

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arXiv2023

On smoothing estimates for Schrödinger equations on product spaces $\mathbb{T}^m\times \mathbb{R}^n$

Let $Δ_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{itΔ_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{α,p}(\mathbb{T}^m\times\mathbb{R}^n)} $$ holds if $p\geq 2(m+n+2)/(m+n)$ and $α> (m+2n)(1/2-1/p)-2/p$. Furthermore, we apply the $\ell^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schrödinger operator $e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}$ in modulation spaces $M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)$: $$ \|e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}f\|_{L^p(\mathbb{T}^m\times\mathbb{R}^n\times [0,1])}\leq C \|f\|_{M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)} $$ for some range of $α$ and $p, q$. The smoothing estimates in $L^p$-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of $p$ and $q$.

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arXiv2022

Pointwise Decay of solutions to the energy critical nonlinear Schrödinger equations

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi \cite{Miyachi} and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space.

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arXiv2022

Scattering for the mass-critical nonlinear Klein-Gordon equations in three and higher dimensions

In this paper we consider the real-valued mass-critical nonlinear Klein-Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method as R. Killip, B. Stovall, and M. Visan [Trans. Amer. Math. Soc. 364 (2012)]. The main new novelty is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.

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arXiv2022

Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schrödinger system on $\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times \mathbb{Z})$. We first establish the variational characterization of the ground state, and derive the threshold of the global well-posedness and scattering. Then we show the global well-posedness and scattering below the threshold by the concentration-compactness/rigidity method, where the almost periodic solution is excluded by adapting the argument in the proof of the mass-critical nonlinear Schrödinger equations by B. Dodson. As a byproduct of the scattering of the cubic focusing infinite coupled nonlinear Schödinger system, we obtain the scattering of the cubic focusing nonlinear Schrödinger equation on the small cylinder, this is the first large data scattering result of the focusing nonlinear Schrödinger equations on the cylinders. In the article, we also show the global well-posedness and scattering of the two dimensional $N-$coupled focusing cubic nonlinear Schrödinger system in $\left(L^2(\mathbb{R}^2) \right)^N$.

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arXiv2021

Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrödinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(Δ_{\mathbb{R}^3 }-x^2 \right) u = |u|^2 u, \quad u|_{t=0} = u_0. \end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson \cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrödinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies.

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arXiv2020

Scattering below the ground state for the 2D non-linear Schrödinger and Klein-Gordon equations revisited

We revisit the scattering problems for the 2D mass super-critical Schrödinger and Klein-Gordon equations with radial data below the ground state in the energy space. We give an alternative proof of energy scattering for both defocusing and focusing cases using the ideas of Dodson-Murphy \citep{dodson2017new-radial}. Our results also include the exponential type nonlinearities which seems to be new for the focusing exponential NLS.

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