Asymptotic stability of the wave equation on compact surfaces and locally distributed damping - A sharp result
M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano
本文研究了具有局部分布阻尼的紧致曲面上的波动方程。我们研究的情况下，阻尼是有效的，在一个精心挑选的子集的任意小的措施。
This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping. We study the case where the damping is effective in a well-chosen subset of arbitrarily small measure.
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In this paper, we study the direct/indirect stability of locally coupled wave equations with local Kelvin-Voigt dampings/damping and by assuming that the supports of the dampings and the coupling coefficients are disjoint. First, we prove the well-posedness, strong stability, and polynomial stability for some one dimensional coupled systems. Moreover, under some geometric control condition, we prove the well-posedness and strong stability in the multi-dimensional case.
Uniform Stabilization of the wave equation on compact surfaces and locally distributed damping
本文研究了具有局部分布阻尼的紧致曲面上的波动方程。我们研究的情况下，阻尼是有效的补充可见脐带套。
We consider the initial value problem for the semilinear wave equation with time-dependent effective damping. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the damping as $t\to \infty$. The result of this paper is the sharp lifespan estimates of blowup solutions for general time-dependent damping including threshold cases between effective and overdamping.
We study a nonlocal evolution equation modeling the deformation of a bridge, either a footbridge or a suspension bridge. Contrarily to the previous literature we prove the asymptotic stability of the considered model with a minimum amount of damping which represents less cost of material. The result is also numerically proved.
We consider a coupled wave system with partial Kelvin-Voigt damping in the interval (-1,1), where one wave is dissipative and the other does not. When the damping is effective in the whole domain (-1,1) it was proven in H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20. That the energy is decreasing over the time with a rate equal to $t^{-\frac{1}{2}}$. In this paper, using the frequency domain method we show the effect of the coupling and the non smoothness of the damping coefficient on the energy de…
Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping
本文研究具有局部分布Kelvin-Voigt反馈的透射板方程能量衰减的渐近性态。确切地说，我们将证明能量至少随时间以几何学的方式衰减。这种方法的独创性来自于这样一个事实，即使用Carleman估计的传输二阶系统，这将是来自板方程建立一个预解估计，提供了著名的Burq的结果[Bur 98]，衰减上述类型。
We consider the Klein-Gordon system posed in an inhomogeneous medium with smooth boundary subject to a local viscoelastic damping distributed around a neighborhoodof the boundary according to the Geometric Control Condition. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gerard . By using sharp Carleman estimates we prove a unique continuation property for coupled syst…
Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$
本文研究了 $\R^2$ 中带临界时间依赖阻尼的可压缩 Euler 系统光滑解的长时间存在性，通过引入向量场 $\widehat{Z}$ 替代传统向量场 $Z$ 以改善线性误差项衰减，建立了关于初始扰动小参数的下界精确寿命估计，该思想亦适用于非线性波动方程的长时间行为研究。
In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler- Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on t…
Strict Lyapunov function for the one-dimensional linear wave equation with a locally distributed damping
针对带局部分布阻尼和狄利克雷边界条件的一维线性波动方程，构造显式严格李雅普诺夫函数。
Sharp higher-order $L^2$-asymptotic expansion of solutions to σ-evolution equations with different damping types
本文在L²框架下研究具有不同阻尼类型的σ-演化方程解的高阶渐近展开，分析类抛物模型（σ₁∈[0,σ/2)）和类σ-演化模型（σ₂∈(σ/2,σ]）对解渐近行为的影响。
Weakly elliptic damping gives sharp decay
本文证明弱椭圆阻尼在非函数演算情形下为抽象阻尼波半群提供尖锐能量衰减，并应用于线性化水波和Kelvin--Voigt阻尼。
This paper is concerned with locally damped semilinear wave equations defined on compact Riemannian manifolds with boundary. We present a construction of measure-controlled damping regions which are sharp in the sense that their summed interior and boundary measures are arbitrarily small. The construction of this class of open sets is purely geometric and allows us to prove a new observability inequality in terms of potential energy rather than the usual one with kinetic energy. A unique continuation property is also proved. Then, in three-dimension spaces, we establish the existence of finite…
In this work, we consider a system of multidimensional wave equations coupled by velocities with one localized fractional boundary damping. First, using a general criteria of Arendt- Batty, by assuming that the boundary control region satisfy some geometric conditions, under the equality speed propagation and the coupling parameter of the two equations is small enough, we show the strong stability of our system in the absence of the compactness of the resolvent. Our system is not uniformly stable in general since it is the case of the interval. Hence, we look for a polynomial decay rate for sm…
We consider the following semilinear wave equation with time-dependent damping. \begin{align} \tag{NLDW} \left\{ \begin{array}{ll} \partial_t^2 u - Δu + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times \mathbb{R}^n, \\ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in \mathbb{R}^n, \end{array} \right. \end{align} where $n \in \mathbb{N}$, $p>1$, $\varepsilon>0$, and $b(t)\thickapprox (t+1)^{-β}$ with $β\in [-1,1)$. It is known that small data blow-up occurs when $1<p< p_F$ and, on the other hand, small data global existence holds when $p>p_F$, where $p_F:=1+2/n$ is the Fujita…
Sharp polynomial decay for polynomially singular damping on the torus
本文研究环面上无界奇异阻尼的波动方程能量衰减，证明多项式控制奇异阻尼具有锐多项式衰减，并进一步给出紧流形上正规 $L^p$-阻尼的多项式衰减及圆周情形的指数衰减结果。
Sharp estimates for the number of degrees of freedom for the damped-driven 2D Navier--Stokes equations
本文导出了二维Navier-Stokes系统和带阻尼Navier-Stokes系统的渐近自由度（决定模态和节点）个数的上界。在第一种情况下，我们得到了以前已知的估计在一个明确的形式，这是大于分形维数的全球吸引。然而，对于具有阻尼的Navier-Stokes系统，我们对决定模式和节点的数量的估计与对全局吸引子的分形维数的尖锐估计相当。我们的调查阻尼驱动的二维Navier-Stokes系统的灵感来自Stommel-Charney正压模式的海洋环流的阻尼代表瑞利摩擦。我们注意到，我们的结果同样适用于Stommel-Charney模型。
We give an elementary proof of sharp decay rates and the linear inviscid damping near monotone shear flow in a periodic channel, first obtained in [14]. We shall also obtain the precise asymptotics of the solutions, measured in the space $L^{\infty}$.
Sharp energy decay rates for the damped wave equation on the torus via non-polynomial derivative bound conditions
本文研究了在紧致流形上阻尼波方程的能量衰减率，通过非多项式导数界条件获得更一般的衰减率，并展示了这些速率如何依赖于阻尼支持集的几何特性。
We consider a class of linear second order differential equations with damping and external force. We investigate the link between a uniform bound on the forcing term and the corresponding ultimate bound on the velocity of solutions, and we study the dependence of that bound on the damping and on the "elastic force". We prove three results. First of all, in a rather general setting we show that different notions of bound are actually equivalent. Then we compute the optimal constants in the scalar case. Finally, we extend the results of the scalar case to abstract dissipative wave-type equation…
This paper is devoted to Bresse systems, a robust model for circular beams, given by a set of three coupled wave equations. The main objective is to establish the existence of global attractors for dynamics of semilinear problems with localized damping. In order to deal with localized damping a unique continuation property (UCP) is needed. Therefore we also provide a suitable UCP for Bresse systems. Our strategy is to set the problem in a Riemannian geometry framework and see the system as a single equation with different Riemann metrics. Then we perform Carleman-type estimates to get our resu…
Sharp conditions for exponential and non-exponential uniform stabilization of the time-dependent damped wave equation
本文研究时变阻尼波动方程的一致镇定问题，证明推广的几何控制条件等价于指数镇定，并对不满足该条件的阻尼给出非指数衰减速率的上下界。
We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. We show that if the damping vanishes as a Hölder function $|x|^β$, and in addition, the boundary of the damped region is strictly convex, the wave is stable at rate $t^{-1+\frac{2}{2β+7}}$, which is better than the known optimal decay rate $t^{-1+\frac{1}{β+3}}$ for strip-shaped dampings of the same Hölder regularity. Moreover, we show by example that the decay rate is optimal. This illustrates the fact that the energy decay rate depends not only on the order o…
Sharp $L^1$-convergence rates to the Barenblatt solutions for the compressible Euler equations with time-varying damping
该研究针对带时变阻尼的可压缩Euler方程，通过建立密度函数与多孔介质方程Barenblatt解的新关联，首次得到适用于任意绝热指数和阻尼参数的统一最优$L^1$收敛速率，显著改进了已有结果。
Energy decay for a locally undamped wave equation
研究了一类阻尼波动方程在几何控制条件不满足的情况下解的能量衰减率。我们假设无阻尼轨迹的集合是一个正余维的平坦环面，并且度量在这个集合周围局部平坦。我们进一步假设阻尼函数在无阻尼集附近的遍历方向上局部具有规定的齐次性。我们证明了一个尖锐的衰减估计在多项式速率依赖于阻尼函数的均匀性。我们的方法依赖于一个精致的microlocal分析连接到第二个microlocalization程序切割的相空间成微小的区域尊重的不确定性原则，但太小的方式进入一个标准的半经典分析本地化。使用乘子方法，我们得到的能量估计在每个区域，然后我们修补的微局部估计在一起。
We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.
We investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping of Kelvin-Voigt type. The main novelty in this paper is that both the damping and the coupling coefficients are non smooth. First, using a general criteria of Arendt-Batty, combined with an uniqueness result, we prove that our system is strongly stable. Next, using a spectrum approach, we prove the non-exponential (uniform) stability of the system. Finally, using a frequency domain approach, combined with a piecewise multiplier technique and the construction of a new multiplier sati…
In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the $L^2$-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the $L^2$-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respective…
We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^β$ near the boundary of the support and show decay at rate $1/t^{\frac{β+2}{β+3}}$. In the case where $W$ vanishes exactly like $x^β$ this result is optimal by work of the second author. The proof uses a version of the Morawetz multiplier method.
We characterize the $L^2$ decay rate of solutions to the 3D magneto-micropolar system in terms of the decay character of the initial datum. Due to a linear damping term, the micro-rotational field has a faster decay rate. We also address the asymptotic behaviour of solutions by comparing them to solutions to the linear part. As a result of the linear damping, the difference between the micro-rotational field and its linear part also decays faster. As part of the proofs of these results, we prove estimates for the derivatives of solutions which might be of independent interest.
Sharp Stability of a String with Local Degenerate Kelvin-Voigt Damping
本文研究了具有局部退化凯尔文-沃伊格阻尼的弹性弦方程的渐进行为，针对不同α值，推导出更优的多项式衰减率，并与最优衰减率一致。
In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of solutions influenced by different damping mechanisms. We show which damping in the model has a dominant influence on smoothing effect, energy estimates, $L^p-L^q$ estimates not necessary on the conjugate line, and on diffusion phenomena. Moreover, we derive asymptotic profiles of solutions in a framework of weighted $L^1$ data. In particular, sharp decay estimates…
Sharp Damping Location for Stability of a Locally Damped Wave Equation and its Galerkin Approximation
研究矩形区域上带垂直阻尼带的变系数波动方程的稳定性，明确阻尼位置与能量衰减的关系，证明其半群的指数稳定性条件，并分析其谱Galerkin近似的收敛性与非一致指数稳定性。
We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.
We study the decay of global energy for the wave equation with Hölder continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp $t^{-1/2}$-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.
This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $\mathbb{R}^3$ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our contribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we sho…
Asymptotic stability of the $2D$ temperature-dependent tropical climate model with the sharp decay rates
本文研究二维温度依赖热带气候模型在正压模态有无线性阻尼两种情形下的渐近稳定性，证明小数据光滑解全局存在，并给出任意 Sobolev 范数下的尖锐时间衰减估计。
Infinite energy solutions for critical wave equation with fractional damping in unbounded domains
本文研究了半线性波动方程在具有分数阶阻尼形式$(-Δ_x+1)^\frac{1}{2}\partial_t u$的$\mathbb{R}^3$无界光滑区域上的无限能量解。推广了已有的有限能量有界域的结果。进一步，在对非线性条件限制较少的情况下，得到了临界五次非线性系统的适定性和局部紧光滑吸引子的存在性，放宽了以往人为的技术条件.这是通过新型的李雅普诺夫泛函，允许建立额外的时空规律性的解决方案的Hohartz型。
Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping
该论文通过双曲尺度变换和局部存在性理论，证明了带非线性阻尼的半线性波动方程在大振幅下解的最大存在时间具有与上界同阶的尖锐下界，确定了指数 $\vartheta(p,q)=\min\{(p-1)/2,(p-q)/q\}$。
In this paper, we consider the longitudinal and transversal vibrations of the transmission Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam and only in one side of the transmission point. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is polynomially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method an…
In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin-Voigt damping, under fully Dirichlet or mixed boundary conditions. Unlike the results of H. L. Zhao, K. S. Liu, and C. G. Zhang and of X. Tian and Q. Zhang, in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin-Voigt damping. Indeed, we prove that the energy of the system decays polynomially and that the obtained decay rate is in some sense optimal. The method is based on the frequency domain approach combining with multiplier method.
In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $t^{-\frac{1}{2}}$ if the two waves have the same speed of propagation, and with rate $t^{-\frac{1}{3}}$ if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order $t^{-1}$.
Nonlinear Landau damping and wave operators in sharp Gevrey spaces
本文在最优加权 Gevrey-3 空间中证明受限 Vlasov–Poisson 系统小扰动解的非线性 Landau 阻尼及相应非线性散射算子的存在性、单射性与 Lipschitz 估计，解决了两个公开问题。
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial dat…
The purpose of this work is to investigate the exponential stability of a second order coupled wave equations by laplacian with one locally internal viscous damping. Firstly, using a unique continuation theorem combined with a Carleman estimate, we prove that our system is strongly stable without any geometric condition. Secondly, using a combination of the multiplier techniques and the frequency domain approach, we show that our system is exponentially stable under \textbf{(PMGC)} condition on the damping region without any restriction on wave propagation speed (i.e whether the two wave equat…
In this paper we consider elastic waves with Kelvin-Voigt damping in 2D. For the linear problem, applying pointwise estimates of the partial Fourier transform of solutions in the Fourier space and asymptotic expansions of eigenvalues and their eigenprojections, we obtain sharp energy decay estimates with additional $L^m$ regularity and $L^p-L^q$ estimates on the conjugate line. Furthermore, we derive asymptotic profiles of solutions under different assumptions of initial data. For the semilinear problem, we use the derived $L^2-L^2$ estimates with additional $L^m$ regularity to prove global (i…
In the present paper, we are concerned with the semilinear viscoelastic wave equation subject to a locally distributed dissipative effect of Kelvin-Voigt type, posed on a bounded domain with smooth boundary. We begin with an auxiliary problem and we show that its solution decays exponentially in the weak phase space. The method of proof combines an observability inequality and unique continuation properties. Then, passing to the limit, we recover the original model and prove its global existence as well as the exponential stability.
We consider the gravity-capillary water waves problem in a domain $Ω_t \subset \mathbb{T} \times \mathbb{R}$ with substantial geometric features. Namely, we consider a variable bottom, smooth obstacles in the flow and a constant background current. We utilize a vortex sheet model introduced by Ambrose, et. al. in arXiv:2108.01786. We show that the water waves problem is locally-in-time well-posed in this geometric setting and study the lifespan of solutions. We then add a damping term and derive evolution equations that account for the damper. Ultimately, we show that the same well-posedness a…
This paper is concerned with weighted energy estimates for solutions to wave equation $\partial_t^2u-Δu + a(x)\partial_tu=0$ with space-dependent damping term $a(x)=|x|^{-α}$ $(α\in [0,1))$ in an exterior domain $Ω$ having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polymonials are given and these decay rate are almost sharp, even when the initial data do not have compact support in $Ω$. The crucial idea is to use special solution of $\partial_t u=|x|^αΔu$ including Kummer's confluent hypergeometric functions.
The aim of this paper is to determine the critical exponent for the nonlinear wave equations with damping and potential terms of the scale invariant order, by assuming that these terms satisfy a special relation. We underline that our critical exponent is different from the one for related equations such as the nonlinear wave equation without lower order terms, only with a damping term, and only with a potential term. Moreover, we study the effect of the decaying order of initial data at spatial infinity. In fact, we prove that not only the lower order terms but also the order of the initial d…
Global Uniqueness and Stability in Determining the Damping Coefficient of an Inverse Hyperbolic Problem with Non-Homogeneous Neumann B.C. through an Additional Dirichlet Boundary Trace
考虑$\mathbb{R}^n$中开有界区域$Ω$上的二阶双曲型方程，其中$C^2$ -边界为$Γ=\paΩ=\bar{Γ_0\cupΓ_1}$，$Γ_0\capΓ_1=\emptyset$，在整个边界$Γ$上满足非齐次Neumann边界条件.然后，我们研究的反问题，确定内部阻尼系数的方程的Dirichlet边界跟踪的解决方案的附加测量装置，在一个合适的，明确的子部分$Γ_1$的边界$Γ$，并在一个可计算的时间间隔$T>0$。在互补部分$Γ_0= Γ\backslashΓ_1$，$T>0$的尖锐条件下，并在弱正则性要求的数据下，我们建立了反问题的两个典型结果：（i）唯一性和（ii）稳定性（在$L^2$水平）。后者（ii）是本文的主要结果。我们的证明依赖于三个主要成分：（a）尖锐Carleman估计在$H^1 \times L_2$ -水平的二阶双曲方程\cite{L-T-Z.1} ;（b）相应的隐含连续的可观测性不等式在同一能量水平\cite{L-T-Z.1} ;（c）尖锐的内部和边界正则性理论的二阶双曲方程与Neumann边界数据\cite{L-T.4}，\cite{L-T.5}，\cite{L-T.6}，\cite{Ta.3}。线性唯一性结果的证明（第4节，第5步）还利用了V.Isakov在\cite [Thm.\，第8.2.2页，{Is.2}。
Scattering for defocusing cubic NLS under locally damped strong trapping
研究变系数三次3D非线性defocusing Schrödinger方程在强捕获情形下的散射问题，通过在捕获区域引入局部阻尼项，对$H^{1+ε}$初始数据建立了$H^s$（$0≤s<1$）的整体存在与散射。
In this article, we study energy decay of the damped wave equation on compact Riemannian manifolds where the damping coefficient is anisotropic and modeled by a pseudodifferential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coefficient satisfies an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Furthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the d…
Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus
该研究针对环面上带幂次型非线性阻尼的一维周期非线性薛定谔方程，证明了小阻尼、近齐次初始数据下的全局存在性与渐近收敛性，给出了范数衰减律及收敛率，明确了动能的衰减特性。
Kirchhoff equations with strong damping
我们认为基尔霍夫方程的强阻尼，即摩擦项取决于权力的“弹性”运营商。我们解决当地和全球存在的解决方案，在两个不同的制度，取决于摩擦项的指数。当指数大于大于1/2时，耗散占优势，在仅假设非线性项的退化双曲性和连续性的条件下，我们得到了能量空间中的整体解的存在性.当指数小于1/2时，我们假设严格双曲性，并考虑一个依赖于非线性项的连续模和阻尼指数的相空间。在这个相空间中，我们证明了局部解的存在性，并在初始值足够小的情况下证明了全局解的存在性。我们假设的规则性的初始数据和非线性项是弱于经典的结果与标准阻尼的基尔霍夫方程。证明利用了线性化方程和适当定义的插值空间的一些最近的尖锐结果。
On a system of semilinear damped $σ$-evolution equations with different damping types in the critical case
研究临界曲线上的非线性项幂指数，通过非线性项的连续性模确定保证Sobolev解全局存在或有限时间爆破的尖锐条件，并解决对称模型的公开问题。
A new proof of nonlinear Landau damping for the 3D Vlasov-Poisson system near Poisson equilibrium
本文针对三维弗拉索夫-泊松系统近泊松平衡态的非线性朗道阻尼问题，基于已有基础工作，通过精确衰减估计与新颖分解技术提供简化证明，揭示扰动密度的类自由输运行为，增进对无约束环境中朗道阻尼的理解。
Landau damping in mixed hyperbolic-kinetic systems and thick sprays
本文研究稠密喷雾模型中 Vlasov 方程与正压可压缩 Euler 方程耦合的朗道阻尼现象，证明声波因与近速粒子相互作用而被阻尼或放大，并揭示该机制在双曲守恒律与 Vlasov 方程耦合时自然出现。
Pseudodifferential damping estimates and stability of relaxation shocks
本文提出基于频率依赖伪微分阻尼估计的方法，用于研究松弛激波的稳定性，强调高频谱稳定性和转折点问题的重要性。
The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $α\to0$ and $γ\to0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.
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