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Yanqiu Guo

Publications and source records attributed to Yanqiu Guo.

At least 19 recordsLinked to original sources
arXiv2026

On a structural acoustic model with logarithmic supercritical source terms

In this paper, we study a structural acoustic model consisting of a semilinear wave equation defined on a three-dimensional bounded domain, coupled with a Kirchhoff-Love plate equation acting on a flat portion of the boundary. Primarily for mathematical interest, we impose nonlinear damping terms and logarithmic-type supercritical source terms on the system. We investigate local and global well-posedness, energy decay rates of potential well solutions, and blow-up of solutions under different conditions on the parameters and initial data. The main novelties include the analysis of the interaction between nonlinear damping and logarithmic energy-amplifying source terms, as well as the development of techniques for handling logarithmic source terms within potential well theory. The logarithmic nonlinearities are not homogeneous under scaling, which creates difficulties in studying potential well solutions. The wave-plate coupling through the acoustic pressure also causes technical difficulties in the analysis, especially in the proof of blow-up of weak solutions.

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arXiv2025

Strong and weak solutions to a structural acoustic model with a $C^1$ source term on the plate

In this manuscript, we consider a structural acoustic model consisting of a wave equation defined in a bounded domain $\Omega \subset \mathbb{R}^3$, strongly coupled with a Berger plate equation acting on the flat portion of the boundary of $\Omega$. The system is influenced by an arbitrary $C^1$ nonlinear source term in the plate equation. Using nonlinear semigroup theory and monotone operator theory, we establish the well-posedness of both local strong and weak solutions, along with conditions for global existence. With additional assumptions on the source term, we examine the Nehari manifold and establish the global existence of potential well solutions. Our primary objective is to characterize regimes in which the system remains globally well-posed despite arbitrary growth of the source term and the absence of damping mechanisms to stabilize the dynamics.

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arXiv2025

Analysis of a three-dimensional rapidly rotating convection model without thermal diffusion

We study a three-dimensional rapidly rotating convection model featuring tall columnar structures, in the absence of thermal diffusion. We establish the global existence and uniqueness of weak solutions, as well as the Hadamard well-posedness of global strong solutions to this model. The lack of thermal diffusion introduces significant challenges in the analysis. To overcome these challenges, we first investigate the regularized model with thermal diffusion and establish delicate estimates that are independent of the thermal diffusion coefficient, and consequently justify the vanishing diffusivity limit. This work serves as a continuation of our previous paper [6].

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arXiv2025

On the two-dimensional Navier-Stokes equations with horizontal viscosity

This paper is concerned with a 2D channel flow that is periodic horizontally but bounded above and below by hard walls. We assume the presence of horizontal viscosity only. We study the well-posedness, large-time behavior, and stability of solutions. For global well-posedness, we aim to assume less differentiability on initial velocity $(u_0, v_0)$: in particular, we assume $u_0,v_0\in L^2(\Omega)$ and $\partial_y u_0 \in L^2(\Omega)$.

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arXiv2025

Vanishing Vertical Viscosity in Two-Dimensional Anisotropic Navier-Stokes Equations with No-Slip Boundary Conditions: An $L^p$ result

This paper studies the inviscid limit problem for the two-dimensional Navier-Stokes equations with anisotropic viscosity. The fluid is assumed to be bounded above and below by impenetrable walls, with a no-slip boundary condition imposed on the bottom wall. For $H^2$ initial velocity, we establish strong convergence in the $L^p$ norm to the limiting problem as the vertical viscosity approaches zero, for any $2\leq p <\infty$. The main challenge lies in the mismatch of boundary conditions - specifically, the no-slip condition in the original problem versus the slip condition in the limiting problem.

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arXiv2024

Sparse distribution of lattice points in annular regions

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of $λ$ and $μ$, where $μ\geq C \log λ$, such that intervals $[λ, \,λ+ μ]$ do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in $\mathbb R^2$ that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in $\mathbb R^2$. Specifically, we establish the existence of annuli $\{x\in \mathbb R^2: λ\leq |x|^2 \leq λ+ κ\}$ with arbitrarily large $λ$ and $κ\geq C λ^s$ for $0<s<\frac{1}{4}$, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold $s=\frac{1}{4}$. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in $\mathbb R^3$.

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arXiv2024

Inertial manifolds for the two-dimensional hyperviscous Navier-Stokes equations

This study establishes the existence of inertial manifolds for the hyperviscous Navier-Stokes equations (HNSE) on a 2D periodic domain: \begin{equation*} \partial_t u+ ν(-Δ) ^βu+(u\cdot \nabla )u+\nabla p=f, \;\; \text{on} \;\; \mathbb{T}^2, \end{equation*} with $\nabla \cdot u=0$, for any $β> \frac{17}{12} $. The exponent $β= \frac{3}{2}$ is identified as the "critical" value for the inertial manifold problem in 2D HNSE, below which the spectral gap condition is not satisfied. A breakthrough in this work is that it extends the theory to "supercritical" regimes where $β< \frac{3}{2}$. An important aspect of our argument involves a refined analysis on the sparse distribution of lattice points in annular regions.

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arXiv2023

Blow-up of a structural acoustics model

This article studies the finite time blow-up of weak solutions to a structural acoustics model consisting of a semilinear wave equation defined on a bounded domain $Ω\subset\mathbb{R}^3$ which is strongly coupled with a Berger plate equation acting on the elastic wall, namely, a flat portion of the boundary. The system is influenced by several competing forces, including boundary and interior source and damping terms. We stress that the power-type source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from $H^1$ into $L^2$. In this paper, we prove the blow-up results for weak solutions when the source terms are stronger than damping terms, by considering two scenarios of the initial data: (i) the initial total energy is negative; (ii) the initial total energy is positive but small, while the initial quadratic energy is sufficiently large. The most significant challenge in this work arises from the coupling of the wave and plate equations on the elastic wall.

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arXiv2022

On the asymptotic behavior of solutions to a structure acoustics model

This article concerns the long term behavior of solutions to a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain $Ω\subset\mathbb{R}^3$ which is coupled with a Berger plate equation acting on a flat portion of the boundary of $Ω$. The system is influenced by several competing forces, in particular a source term acting on the wave equation which is allowed to have a supercritical exponent. Our results build upon those obtained by Becklin and Rammaha [8]. With some restrictions on the parameters in the system and with careful analysis involving the Nehari manifold we obtain global existence of potential well solutions and establish either exponential or algebraic decay rates of energy, dependent upon the behavior of the damping terms. The main novelty in this work lies in our stabilization estimate, which notably does not generate lower-order terms. Consequently, the proof of the main result is shorter and more concise.

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arXiv2020

Global well-posedness for a rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

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arXiv2018

Global well-posedness for nonlinear wave equations with supercritical source and damping terms

We prove the global well-posedness of weak solutions for nonlinear wave equations with supercritical source and damping terms on a three-dimensional torus $\mathbb T^3$ of the prototype \begin{align*} &u_{tt}-Δu+|u_t|^{m-1}u_t=|u|^{p-1}u, \;\; (x,t) \in \mathbb T^3 \times \mathbb R^+ ; \notag\\ &u(0)=u_0 \in H^1(\mathbb T^3)\cap L^{m+1}(\mathbb T^3), \;\; u_t(0)=u_1\in L^2(\mathbb T^3), \end{align*} where $1\leq p\leq \min\{ \frac{2}{3} m + \frac{5}{3} , m \}$. Notably, $p$ is allowed to be larger than $6$.

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arXiv2018

Global regularity for a rapidly rotating constrained convection model of tall columnar structure with weak dissipation

We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term.

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arXiv2018

Energy decay of a viscoelastic wave equation with supercritical nonlinearities

This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: \begin{align*} \begin{cases} u_{tt}- k(0) Δu - \int_0^{\infty} k'(s) Δu(t-s) ds +|u_t|^{m-1}u_t =|u|^{p-1}u, \quad \text{ in } Ω\times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } Ω\times (-\infty,0], \end{cases} \end{align*} where $Ω$ is a bounded domain in $\mathbb R^3$ with a Dirichlét boundary condition and $u_0$ represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified provided that the history value $u_0$ is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel $-k'(s)$ as well as the growth rate of the damping term. This manuscript complements our previous work [Guo et al. in J Differ Equ 257, 3778-3812(2014), J Differ Equ 262, 1956-1979(2017)] where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to $-\infty$.

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arXiv2016

Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: \begin{align*} \begin{cases} u_{tt}- k(0) Δu - \int_0^{\infty} k'(s) Δu(t-s) ds + |u_t|^{m-1}u_t=|u|^{p-1}u, \;\;\;\;\; Ω\times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } Ω\times (-\infty,0], \end{cases} \end{align*} where $Ω$ is a bounded domain in $\mathbb R^3$ with a Dirichlét boundary condition. The relaxation kernel $k$ is monotone decreasing and $k(\infty)=1$. We study blow-up of solutions when the source is stronger than dissipations, i.e., $p> \max\{m,\sqrt{k(0)}\}$, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to $-\infty$, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.

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arXiv2016

Non-viscous Regularization of the Davey-Stewartson Equations: Analysis and Modulation Theory

In the present study we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in variety of parameter regimes under which the finite blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the $α$-models of turbulence and therefore are called the $α$-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the $α$-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work of Cao, Musslimani and Titi on the non-viscous $α$-regularization of the nonlinear Schrödinger equation.

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arXiv2015

On the backward behavior of some dissipative evolution equations

We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the periodic 2D Navier-Stokes equations studied by Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we study the backward behavior of solutions to the damped driven nonlinear Schrodinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, we provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, we discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by the investigation of the Bardos-Tartar conjecture stated in [5].

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arXiv2014

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

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