Searcharxiv⌕ Search

arXiv subjects

Yuming Qin

Publications and source records attributed to Yuming Qin.

At least 19 recordsLinked to original sources
arXiv2025

Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory

This paper delves into the long-time dynamics of a non-autonomous viscoelastic Kirchhoff plate equation with memory effects, described by $$ u_{t t}-\Delta u_{t t}+a_\epsilon(t) u_t+\alpha \Delta^2 u-\int_0^{\infty} \mu(s) \Delta^2 u(t-s) \mathrm{d} s-\Delta u_t+f(u)=g(x,t), $$ in bounded domain $\Omega \subset \mathbb{R}^N$ with smooth boundary and nonlinear terms. Initially, the global existence of a weak solution that induces a continuous process is established. Subsequently, the existence of a uniform attractor is demonstrated in both subcritical and critical growth scenarios, utilizing operator techniques and an innovative analytical approach. Finally, the upper semicontinuity of the family of uniform attractors as the pert parameterurbation $\epsilon \to 0^+$ is proven through delicate energy estimates and a contradiction argument. Our results not only extend classical attractor theory to more general non-autonomous viscoelastic systems but also resolve open questions regarding the limiting behavior of attractors in the presence of both memory and critical nonlinearity.

math.AP↗
arXiv2025

Local-in-time well-posedness for 2D compressible magneto-micropolar boundary layer in Sobolev spaces

In this paper, we study the two-dimensional compressible magneto-micropolar boundary layer equations on the half-plane, which are derived from 2D compressible magneto-micropolar fluid equations with the non-slip boundary condition on velocity, Dirichlet boundary condition on micro-rotational velocity and perfectly conducting boundary condition on magnetic field. Based on a nonlinear coordinate transformation proposed in \cite{LXY2019}, we first prove the local-in-time well-posedness for the compressible magneto-micropolar boundary layer system in Sobolev spaces, provided that initial tangential magnetic field is non-degenerate.

math.AP↗
arXiv2024

Uniform attractor of a non-autonomous Lame thermoelastic system

In this paper, we investigate the dynamical behavior of non-autonomous Lame thermoelastic systems within $N$-dimensional materials. With appropriate constraints on nonlinear characteristics and functional parameters, we initially establish the existence of a uniformly absorbing set by constructing a Lyapunov function. Subsequently, we employ the contraction mapping principle to demonstrate the uniformly asymptotic compactness of the system. Finally, under irrotational conditions, we prove the existence of a uniform attractor $\mathcal{A}_\Sigma$ in the space $H_c$.

math.AP↗
arXiv2024

Pullback attractors for nonclassical diffusion equations with a delay operator

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical non-autonomous diffusion equations with a delay operator in time-dependent spaces when the nonlinear function $g$ satisfies subcritical exponent growth conditions, the delay operator $\varphi(t, u_t)$ contains some hereditary characteristics and the external force $k \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(\Omega)\right)$. First, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{\mathcal{H}_{t}(\Omega)}$ and $C_{\mathcal{H}^{1}_{t}(\Omega)}$, respectively.

math.AP↗
arXiv2024

Prandtl Equations and Related Boundary Layer Equations

This book aims to present some recent results on Prandtl equations and MHD boundary layer equations. This book is essentially divided into two parts. Chapter 1 as the first part systematically surveys the results till 2020 on Prandtl equations and MHD boundary layer equations. Chapter 2 to 6 are the main part of the book, which presents the local and the global well-posedness of solutions to the Prandtl equations and MHD boundary layer equations. In detail, Chapter 2 is concerned with global well-posedness of solutions to the 2D Prandtl-Hartmann equations in an analytic framework. Chapter 3 investigates the local existence of solutions to the 2D Prandtl equations in a weighted Sobolev space. Chapter 4 studies the local well-posedness of solutions to the 2D mixed Prandtl equations in a Sobolev space without monotonicity and lower bound. Chapter 5 is concerned with global existence of solutions to the 2D magnetic Prandtl equations in the Prandtl-Hartmann regime. Chapter 6 proves the local existence of solutions to the 3D Prandtl equations with a special structure. Mathematicians and physicists who are interested in fluid dynamics will find this book helpful.

math.AP↗
arXiv2024

Long time well-posedness for the 3D Prandtl boundary layer equations with a special structure

This paper is concerned with existence, uniqueness and stability of the solution for the 3D Prandtl equation in a polynomial weighted Sobolev space. The main novelty of this paper is to directly prove the long time well-posedness to 3D Prandtl equation under monotonicity condition $\partial_{z} u >0$ and a special structural assumption $v=Ku$ $\big(\partial_{z}\big(\frac{v}{u}\big) \equiv 0\big)$ by the energy method. Moreover, the solution's lifespan can be extended to any large $T$, provided that the initial data with a perturbation lie in the monotonic shear profile of small size $e^{-T}$. This result extends the local well-posedness results established by Liu-Wang-Yang \cite{Liu-Wang-Yang-1-2017} (Adv. Math. 308 (2017) 1074-1126) and Qin-Wang \cite{Qin-Wang-2024} (J. Math. Pure. Appl. 194 (2025) 103670) for the 3D Prandtl equations to long-time well-posedness.

math.AP↗
arXiv2024

Asymptotic behavior for the fast diffusion equation with absorption and singularity

This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of $u_t=\triangle u^m -u^p$. We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies $0 1$. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\frac{n-1}{n} m+\frac{2}{n}$ via the entropy dissipation method combining the generalized Shannon's inequality and Csisz$\mathrm{\acute{a}}$r-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.

math.AP↗
arXiv2024

Existence and invariant measure of pullback attractors for 3D Navier-Stokes-Voigt equations with delay

In this paper, we study the long-time dynamics of 3D non-autonomous Navier-Stokes-Voigt(NSV) equations with delay. Inspired by [36], we use the contractive function method to prove the pullback D-asymptotical compactness and existence of the pullback attractors. Furthermore, we verify the regularity of pullback attractors by the method in [14, 43, 47] and there exists a unique family of Borel invariant probability measures which is supported by the pullback attractors.

math.AP↗
arXiv2024

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

In this paper, we shall investigate the existence and upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with a strong damping in the time-dependent space $X_t$. After deriving the existence and uniqueness of solutions by the Faedo-Galerkin approximation method, we establish the existence of pullback attractors. Later on, we prove the upper semicontinuity of pullback attractors between the Kirchhoff-type wave equations with $δ\geq 0$ and the conventional wave equations with $δ=0$ by a series of complex energy estimates.

math.AP↗
arXiv2023

Existence and regularity of pullback attractors for nonclassical non-autonomous diffusion equations with delay

In this paper, we consider the asymptotic behavior of weak solutions for non-autonomous diffusion equations with delay in time-dependent spaces when the nonlinear function $f$ is critical growth, the delay term $g(t, u_t)$ contains some hereditary characteristics and the external force $h \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(Ω)\right)$. Firstly, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{\mathcal{H}_{t}(Ω)}$ and $C_{\mathcal{H}^{1}_{t}(Ω)}$ respectively.

math.AP↗
arXiv2023

Local existence of solutions to 3D Prandtl equations with a special structure

In this paper, we consider the 3D Prandtl equation in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the Crocco transform has always been used with the general outer flow $U\neq\text{constant}$, this Crocco transform is not needed here for 3D Prandtl equations. We use the skill of cancellation mechanism and construct a new unknown function to show that the existence and uniqueness of solutions to 3D Prandtl equations (cf. Masmoudi and Wong, Comm. Pure Appl. Math., 68(10)(2015), 1683-1741) which extends from the two dimensional case in \cite {12} to the present three dimensional case with a special structure.

math.AP↗
arXiv2023

Stability of a one-dimensional full viscous quantum hydrodynamic system

A full viscous quantum hydrodynamic system for particle density, current density, energy density and electrostatic potential coupled with a Poisson equation in one dimensional bounded intervals is studied. First, the existence and uniqueness of a steady-state solution to the quantum hydrodynamic system is established. Then, utilizing the fact that the third order perturbation term has an appropriate sign, the local-in-time existence of the solution is investigated by introducing a fourth order viscous regularization and using the entropy dissipation method. In the end, the exponential stability of the steady-state solution is shown by constructing a uniform a-priori estimate.

math.AP↗
arXiv2023

Strong global attractors for a three dimensional nonclassical diffusion equation with memory

In this paper, we study the strong global attractors for a three dimensional nonclassical diffusion equation with memory. First, we prove the existence and uniqueness of strong solutions for the equations by the Galerkin method. Then we prove the existence of global attractors for the equations in $H^2(Ω)\cap H^1_0(Ω)\times L^2_μ(\mathbb{R}^+;H^2(Ω)\cap H^1_0(Ω))$ by the condition (C).

math.AP↗
arXiv2023

Strong attractors for the nonclassical diffusion equation with fading memory in time-dependent spaces

In this paper, we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term $f$ fulfills the polynomial growth of arbitrary order and the external force $ g(x)\in L^{2}(Ω)$. In the framework of time-dependent spaces, we verify the existence and uniqueness of strong solutions by the Galerkin method, then we obtain the existence of the time-dependent global attractor $\mathscr{A}=\{A_t\}_{t\in \mathbb{R}}$ in $\mathcal{M}_t^1$.

math.AP↗
arXiv2023

Regularity of pullback attractors for nonclassical diffusion equations with delay

In this paper, we mainly study the regularity of pullback $\mathcal{D}$-attractors for a nonautonomous nonclassical diffusion equation with delay term $b(t,u_t)$ which contains some hereditary characteristics. Under a critical nonlinearity $f$, a time-dependent force $g(t,x)$ with exponential growth and a delayed force term $b(t,u_t)$, we prove that there exists a pullback $\mathcal{D}$-attractor $\mathcal{A}=\{A(t):t \in \mathbb{R}\}$ in $\mathbb{K}^1=H_0^1(Ω) \times L^2((-h,0);L^2(Ω))$ to problem \eqref{ine01} and for each $t \in \mathbb{R}$, $A(t)$ is bounded in $\mathbb{K}^2=H^2(Ω) \cap H_0^1(Ω) \times L^2((-h,0);L^2(Ω))$.

math.AP↗
arXiv2023

Existence and regularity of global attractors for a Kirchhoff wave equation with strong damping and memory

This paper is concerned with the existence and regularity of global attractor $\mathcal A$ for a Kirchhoff wave equation with strong damping and memory in the weighted time-dependent spaces $\mathcal H$ and $\mathcal H^{1}$, respectively. In order to obtain the existence of $\mathcal A$, we mainly use the energy method in the priori estimations, and then verify the asymptotic compactness of the semigroup by the method of contraction function. Finally, by decomposing the weak solutions into two parts and some elaborate calculations, we prove the regularity of $\mathcal A$.

math.AP↗
arXiv2022

Existence and regularity of pullback attractors for a non-autonomous diffusion equation with delay and nonlocal diffusion in time-dependent spaces

In this paper, we study the asymptotic behavior of solution to a non-autonomous diffusion equations with delay containing some hereditary characteristics and nonlocal diffusion in time-dependent space $C_{\mathcal{H}_{t}(Ω)}$. When the nonlinear function $f$ satisfies the polynomial growth of arbitrary order $p-1$ $(p \ge 2)$ and the external force $h \in L_{l o c}^{2}\left(\mathbb{R} ; H^{-1}(Ω)\right)$, we establish the existence and regularity of the time-dependent pullback attractors.

math.AP↗