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Guilong Gui

Publications and source records attributed to Guilong Gui.

17 recordsLinked to original sources
arXiv2026

Global well-posedness of 2-D incompressible anisitropic Navier-Stokes equations with variable density

We establish the global well-posedness for two-dimensional inhomogeneous, incompressible, anisotropic Navier-Stokes systems. Two specific models are analyzed: one with partial dissipation (referred to as (AINS)) and one with only horizontal dissipation (referred to as (HINS)), under the assumption that the initial density is bounded away from zero and infinity. For the (AINS) system posed in the whole plane $\mathbb{R}^2$, we prove the existence and uniqueness of global solutions for finite-energy initial data, employing time-weighted energy estimates and a duality argument. For the (HINS) system on the domain $\mathbb{T} \times \mathbb{R}$, global well-posedness is established for sufficiently small initial velocity and sufficiently small density variation. By exploiting the anisotropic dissipation structure, employing Poincar\'{e}-type inequalities to obtain exponential decay for the oscillatory part of the velocity field, and carefully balancing the growth of the density gradient, we overcome the principal analytical challenges.

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arXiv2025

Global well-posedness of 3-D density-dependent incompressible MHD equations with variable resistivity

In this paper, we investigate the global existence of weak solutions to 3-D inhomogeneous incompressible MHD equations with variable viscosity and resistivity, which is sufficiently close to $1$ in $L^\infty(\mathbb{R}^3),$ provided that the initial density is bounded from above and below by positive constants, and both the initial velocity and magnetic field are small enough in the critical space $\dot{H}^{\frac{1}{2}}(\mathbb{R}^3).$ Furthermore, if we assume in addition that the kinematic viscosity equals $1,$ and both the initial velocity and magnetic field belong to $\dot{B}^{\frac{1}{2}}_{2,1}(\mathbb{R}^3),$ we can also prove the uniqueness of such solution.

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arXiv2024

Global refined Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes equations with large density

We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations with initial velocity $u_0$ being sufficiently small in critical spaces and with initial density being bounded from above and below. We first prove the global existence of Fujita-Kato solution to the system if we assume in addition that the initial velocity is in the critical Sobolev space. While under the additional assumptions that the initial velocity is in the critical Besov space and initial density is in a critical Besov space, we prove that the solutions are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system but also improve the exponential-in-time growth estimate for the solution in the paper [Abidi-Gui-Zhang, ARMA 2012] to be the uniform-in-time estimate.

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arXiv2024

Low mach Number Limit of the Viscous and Heat Conductive Flow with general pressure law on torus

We prove the low Mach number limit from compressible Navier-Stokes-Fourier system with the general pressure law around a constant state on the torus $\mathbb{T}^N_a$. We view this limit as a special case of the weakly nonlinear-dissipative approximation of the general hyperbolic-parabolic system with entropy. In particular, we consider the ill-prepared initial data, for which the group of fast acoustic waves is needed to be filtered. This extends the previous works, in particular Danchin [ Amer. J. Math. 124 (2002), 1153-1219] in two ways: 1. We treat the fully general non-isentropic flow, i.e. the pressure depends on the density $ρ$ and temperature $θ$ by basic thermodynamic law. We illustrate the role played by the entropy structure of the system in the coupling of the acoustic waves and incompressible flow, and the construction of the filtering group. 2. We refine the small divisor estimate, which helps us to give the first explicit convergence rate of the filtered acoustic waves whose propogation is governed by non-local averaged system. In previous works, only convergence rate of incompressible limit was obtained.

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arXiv2024

Construction of low regularity strong solutions to the viscous surface wave equations

We construct in the paper the low-regularity strong solutions to the viscous surface wave equations in anisotropic Sobolev spaces. Here we use the Lagrangian structure of the system to homogenize the free boundary conditions, and establish a new iteration scheme on a known equilibrium domain to get the low-regularity strong solutions, in which no nonlinear compatibility conditions on the initial data are required.

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arXiv2023

On the global existence and uniqueness of solution to 2-D inhomogeneous incompressible Navier-Stokes equations in critical spaces

In this paper, we establish the global existence and uniqueness of solution to $2$-D inhomogeneous incompressible Navier-Stokes equations \eqref{1.2} with initial data in the critical spaces. Precisely, under the assumption that the initial velocity $u_0$ in $L^2 \cap\dot B^{-1+\frac{2}{p}}_{p,1}$ and the initial density $ρ_0$ in $L^\infty$ and having a positive lower bound, which satisfies $1-ρ_0^{-1}\in \dot B^{\frac{2}λ}_{λ,2}\cap L^\infty,$ for $p\in[2,\infty[$ and $λ\in [1,\infty[$ with $\frac{1}{2}<\frac{1}{p}+\frac{1}λ\leq1,$ the system \eqref{1.2} has a global solution. The solution is unique if $p=2.$ With additional assumptions on the initial density in case $p>2,$ we can also prove the uniqueness of such solution. In particular, this result improves the previous work in \cite{AG2021} where $u_{0}$ belongs to $\dot{B}_{2,1}^{0}$ and $ρ_0^{-1}-1$ belongs to $\dot{ B}_{\frac{2}{\varepsilon},1}^{\varepsilon}$, and we also remove the assumption that the initial density is close enough to a positive constant in \cite{DW2023} yet with additional regularities on the initial density here.

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arXiv2022

Global stability of the compressible viscous surface waves in an infinite layer

We investigate in this paper the global stability of the compressible viscous surface waves in the absence of surface tension effect with a steady-state violating Rayleigh-Taylor instability and the reference domain being the horizontal infinite layer. The fluid dynamics are governed by the 3-D gravity-driven isentropic compressible Navier-Stokes equations. We develop a mathematical approach to establish global well-posedness of free boundary problems of the multi-dimensional compressible Navier-Stokes system based on the Lagrangian framework, which requires no nonlinear compatibility conditions on the initial data.

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arXiv2019

Lagrangian approach to global well-posedness of the viscous surface wave equations without surface tension

In this paper, we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab, for which the first complete proof was given in Guo-Tice (2013, Analysis and PDE) via a hybrid of Eulerian and Lagrangian schemes. The fluid dynamics are governed by the gravity-driven incompressible Navier-Stokes equations. Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows, few mathematical works for global existence are based on such an approach in the absence of surface tension effect, due to breakdown of Beale's transformation. We develop a mathematical approach to establish global well-posedness based on the Lagrangian framework by analyzing suitable "good unknowns" associated with the problem, which requires no nonlinear compatibility conditions on the initial data.

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arXiv2019

Semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition

In this paper, we justify the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition on the 3-D upper space under the assumption that the leading order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in \cite{CR2009, PNB2005} concerning the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition.

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arXiv2018

Global well-posedness of the 2-D incompressible Navier-Stokes-Cahn-Hilliard system with a singular free energy density

This paper would focus on the subject of the 2-D incompressible Navier-Stokes-Cahn-Hilliard (NS-CH) system with a singular free energy density. Due to lack of the maximum principle for the convective Cahn-Hilliard equation (as a fourth-order parabolic equation), we construct its approximate second-order parabolic equation, and use comparison principle and the basic energy estimates to separate the solution from the singular values of the singular free energy density, where the Orlicz embedding theorem plays a key role. Based on these, we prove the global well-posedness of the Cauchy problem of the 2-D NS-CH equations with periodic domains by using energy estimates and the Logarithmic Sobolev inequality.

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arXiv2018

A nonlocal shallow-water model arising from the full water waves with the Coriolis effect

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

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arXiv2018

On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect

We consider an asymptotic 1D (in space) rotation-Camassa-Holm (R-CH) model, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due to the Earth's rotation. This model equation has similar wave-breaking phenomena as the Camassa-Holm equation. It is analogous to the rotation-Green-Naghdi (R-GN) equations with the weak Earth's rotation effect, modeling the propagation of wave allowing large amplitude in shallow water. We provide here a rigorous justification showing that solutions of the R-GN equations tend to associated solution of the R-CH model equation in the Camassa-Holm regime with the small amplitude and the larger wavelength. Furthermore, we demonstrate that the R-GN model equations are locally well-posed in a Sobolev space by the refined energy estimates.

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arXiv2012

On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity

Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. In addition, the persistence properties of the strong solutions for the equation are obtained.

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arXiv2011

On the wave-breaking phenomena and global existence for the generalized periodic Camassa-Holm equation

Considered herein is the initial-value problem for the generalized periodic Camassa-Holm equation which is related to the Camassa-Holm equation and the Hunter-Saxton equation. Sufficient conditions guaranteeing the development of breaking waves in finite time are demonstrated. On the other hand, the existence of strong permanent waves is established with certain initial profiles depending on the linear dispersive parameter in a range of the Sobolev spaces. Moreover, the admissible global weak solution in the energy space is obtained.

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