Kais Ammari, Mouez Dimassi, Maher Zerzeri
In this paper, we show that the fastest decay rate for some Petrowsky-like dissipative systems is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup, if the corresponding operator satisfied some spectral gap condition. We give also some applications to illustrate our setting.
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In this paper, by using Fourier splitting method and the expanded properties of decay character $r^*$, we establish the algebraic decay rate of higher order derivative of solutions to 2D dissipative quasi-geostrophic flows.
We address the study of decay rates of solutions to dissipative equations. The characterization of these rates is given for a wide class of linear systems by the {\em decay character}, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. We then use the decay character and the Fourier Splitting method to obtain upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case.
We study the global existence and decay estimates for nonlinear wave equations with the space-time dependent dissipative term in an exterior domain. The linear dissipative effect may vanish in a compact space region. Moreover the nonlinear terms need not divergence form. For getting the higher order energy estimates, we introduce an argument using the rescaling. The method is useful to control derivatives of the dissipative coefficient.
This article gives an energy decay result for small data solutions to a class of semilinear wave equations in two space dimensions possessing weakly dissipative structure relevant to the Agemi condition.
We study global existence and decay estimates for quasilinear wave equations with dissipative terms in the Sobolev space $H^L \times H^{L-1}$, where $L \geq [d/2]+3$. The linear dissipative terms depend on space variable coefficient, and these terms may vanish in some compact region. For the proof of global existence, we need estimates of higher order energies. To control derivatives of the dissipative coefficient, we introduce an argument using the rescaling. Furthermore we get the decay estimates with additional assumptions on the initial data. To obtain the decay estimates, the rescaling ar…
On the decay estimate for small solutions to nonlinear Klein-Gordon equations with dissipative structure
本文研究一维三次非线性Klein-Gordon方程Cauchy问题的小数据解，给出其$L^p$-衰减估计，并证明在适当耗散结构下解比自由解衰减更快。
Large-data $L^2$-decay for attractive-dissipative nonlinear Schrödinger equations without the strong dissipative condition
本文基于吸引-耗散非线性薛定谔方程，通过引入增强能量，克服了标准能量在Reλ<0时的符号不定性，从而得到大初始数据的$L^2$衰减估计，无需强耗散条件。
We consider the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution gains an additional logarithmic decay in comparison with the free evolution in the sense of $L^p$, $2\le p \le \infty$.
On the energy decay estimates for the weak dissipative wave equations with oscillating coefficient
本文发展 Ghisi-Gobbino 方法，分析带振荡弱耗散系数的耗散波动方程在接近衰减临界情形时的能量衰减，并考察初始数据光滑性的影响。
On the energy decay estimate for the dissipative wave equation with very fast oscillating coefficient and smooth initial data
本文针对系数含弱耗散与极快振荡项的耗散波动方程柯西问题，通过引入评估振荡抵消的新系数条件并结合Gevrey类光滑初值，证明了能量衰减估计。
Given $A,B\in M_n(\mathbb R)$, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form \begin{equation*} \partial_{t}u+A\partial_{x}u+Bu=0, \end{equation*} with the aim of providing a detailed description of the large-time behavior. Sharp $L^p$-$L^q$ estimates are established for the distance between the solution to the system and a time-asymptotic profile, where the profile is the superposition of diffusion waves and exponentially decaying waves.
In \cite{GW12} (Y. Guo, Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equ. 37 (2012) 2165--2208), Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms.…
We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t -> +infinity, at least as fast as a suitable negative power of t. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t. Our results are stated and proved in an abstract Hilbert space setting, and then applied to …
In this paper, we first generalize a new energy approach, developed by Y. Guo and Y. Wang \cite{GW12}, in the framework of homogeneous Besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation, then we apply this approach to the supercritical and critical quasi-geostrophic equation and the critical Keller-Segel system. We show that the negative Besov norm of solutions is preserved along time evolution, and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the Fourier splitting approach and the interpolat…
In earlier works, we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.
Very recently, a new decay framework has been given by [51] for linearized dissipative hyperbolic systems satisfying the Kawashima-Shizuta condition on the framework of Besov spaces, which allows to pay less attention on the traditional spectral analysis. However, owing to interpolation techniques, the analysis remains valid only for nonlinear systems in higher dimensions $(n\geq3)$ and the corresponding case of low dimensions was left open, which provides the main motivation of this work. Firstly, we develop new time-decay properties on the frequency-localization Duhamel principle. Furthermor…
We consider the Cauchy problem of a dissipative nonlinear Schrödinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a nonlinear power and time decay order of harmonic potential.
We give a new decay framework for general dissipative hyperbolic system and hyperbolic-parabolic composite system, which allow us to pay less attention on the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish the optimal decay estimates on the framework of spati…
We consider a nonlinear model for electrical conduction in biological tissues. The nonlinearity appears in the interface condition prescribed on the cell membrane. The purpose of this paper is proving asymptotic convergence for large times to a periodic solution when time-periodic boundary data are assigned. The novelty here is that we allow the nonlinearity to be noncoercive. We consider both the homogenized and the non-homogenized version of the problem.
We consider the Cauchy problem for systems of semilinear wave equations in two space dimensions. We present a structural condition on the nonlinearity under which the energy decreases to zero as time tends to infinity if the Cauchy data are sufficiently small, smooth and compactly-supported.
We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data is in $L^2$ only, we prove that the $L^2$ norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For the initial data in $L^p \cap L^2$, with $1 \leq p < 2$, we are able to obtain a uniform decay rate in $L^2$. We also prove that when the $L^{\frac{2}{2 α-1}}$ norm of the initial data is small enough, the $L^q$ norms, for $q > \frac{2}{2 α-1}$ have uniform decay rates. This result allows us to prove decay for the $L^q$ norms,…
Refinement of the $L^{2}$-decay estimate of solutions to nonlinear Schrödinger equations with attractive-dissipative nonlinearity
本文改进非线性耗散薛定谔方程解的 $L^{2}$ 衰减估计，将限制放宽至 $p \le 1+4/(3d)$，并在 $p < 1+4/(3d)$ 时建立最佳衰减率。
Sharp well-posedness and spatial decaying for a generalized dispersive-dissipative Kuramoto-type equation and applications to related models
本文引入含分数阶拉普拉斯算子的广义色散-耗散Kuramoto型方程，在负阶Sobolev空间证明尖锐适定性，研究解的最优空间逐点衰减性质，与孤立波数值研究一致。
In this paper, we consider the following dissipative viscoelastic with memory-type Timoshenko system \begin{equation*} \begin{gathered} \begin{cases} ρ_1 ϕ_{tt} - κ( ϕ_{x} + ψ) _x + κ\int_0^\infty g(s) (ϕ_x +ψ)_x(t-s) ~ds =0 & \text{in}~ \left( {0,L} \right) \times \mathbb{R}^+ , \\ ρ_2 ψ_{tt} - b ψ_{xx} + κ( ϕ_{x} + ψ)-κ\int_0^\infty g(s) (ϕ_x +ψ)(t-s)~ ds=0 & \text{in}~ \left( {0,L} \right) \times \mathbb{R}^+ , \\ \end{cases} \end{gathered} \end{equation*} with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some nonnegative functions $ξ$ and $…
Due to the dissipative structure of \textit{regularity-loss}, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions to dissipative systems. The aim of this paper is to seek the lowest regularity index for the optimal decay rate of $L^{1}(\mathbb{R}^n)$-$L^2(\mathbb{R}^n)$. Consequently, a notion of minimal decay regularity for dissipative systems of regularity-loss is firstly proposed. To do this, we develop a new time-decay estimate of $L^p(\mathbb{R}^n)$-$L^{q}(\mathbb{R}^n)$-$L^{r}(\mathbb{R}^n)$ type b…
We study the dissipative linear wave equation in a bounded domain. The exponential decay rate of the energy was established by Bardos, Lebeau and Rauch under a geometrical hypothesis linked with the geodesics. Furthermore such condition called geometric control condition is almost necessary to get a uniform exponential decay. In another hand, Lebeau proved a logarithmic decay rate for smooth solutions when no particular geometric condition is required. In this paper we give for some particular geometries a polynomial decay rate when the geometric control condition is not fulfilled.
This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by conditions equivalent to the uniform decay of all Fourier modes at the reference state. On a technical level, the theory …
We characterize the set of functions $u\_0\in L^2(R^n)$ such that the solution of the problem $u\_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting from $u\_0$ satisfy upper and lower bounds of the form $c(1+t)^{-γ}\le \|u(t)\|\_2\le c'(1+t)^{-γ}$.Here $\mathcal{L}$ is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on $u\_0$ for a solution of the Navier--Stokes system to sati…
We study the critical dissipative quasi-geostrophic equations in $\bR^2$ with arbitrary $H^1$ initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.
Boundedness and decay for the conformal wave equation in Schwarzschild-AdS under dissipative boundary conditions
研究四维Schwarzschild-反de Sitter时空中耗散边界条件下的保形波动方程，证明了在控制n次T-交换能量时，非退化能量以任意多项式速率(1+v)^{-n}衰减，与Dirichlet边界条件下的对数衰减形成对比。
We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them witho…
This paper studies the dissipative structure of the system of equations that describes the motion of a compressible, isothermal, viscous-capillar fluid of Korteweg type in one space dimension. It is shown that the system satisfies the genuine coupling condition of Humpherys (J. Hyperbolic Differ. Equ. 2, 2005, no. 4, 963-974), which is, in turn, an extension to higher order systems of the classical condition by Kawashima and Shizuta (Tohoku Math. J. 40, 1988, no. 3, 449-464; Hokkaido Math. J. 14, 1985, no. 2, 249-275) for second order systems. It is proved that genuine coupling implies the dec…
In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in deriving the global a priori estimates of solutions. We overcome this difficulty by employing a time-weighted $L^2$ energy method which makes use of the integrability of $\|(\p^2_xu_t,\p^3_xu)(t)\|_{L^{\infty}}$. This $L^\infty$ norm can be controlled by showing the optimal $L^2$ decay estimates for lower-order derivatives of solutions. Thus we obtain the desir…
We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Expo- nential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with th…
Exponential decay for the Schr{ö}dinger equation on a dissipative waveguide
针对边界处处吸收却不满足几何控制条件的耗散波导，本文利用变量分离与横向算子特征函数的 Riesz 基性质，证明 Schrödinger 方程解呈指数衰减，并讨论负吸收指数情形。
In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf A:=(A^1,\dots,A^d)\in (\mathbb R^{n\times n})^d$ be a matrix-vector, where $A^j\in\mathbb R^{n\times n}$, and let $B\in \mathbb R^{n\times n}$ be not required to be symmetric but have one single eigenvalue zero, we consider the Cauchy problem for linear $n\times n$ systems having the form \begin{equation*} \partial_{t}u+\mathbf A\cdot \nabla_{\mathbf x} u+Bu=0,\qquad…
This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in \cite{UKS84,SK85}, and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf.~the Timoshenko system and the Euler-Maxwell system). Moreover, it had been al…
In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global …
Sharp decay characterization for partially dissipative hyperbolic systems of balance laws
本文通过L^p能量方法，提出新有效量Ψ(t,x)，揭示部分耗散双曲系统解在dot B^{σ}_{p,1}范数下的衰减速率及耗散部分的增强衰减速率。
Dissipative hyperbolic systems of \textit{regularity-loss} have been recently received increasing attention. Usually, extra higher regularity is assumed to obtain the optimal decay estimates, in comparison with that for the global-in-time existence of solutions. In this paper, we develop a new frequency-localization time-decay property, which enables us to overcome the technical difficulty and improve the minimal decay-regularity for dissipative systems. As an application, it is shown that the optimal decay rate of $L^1(\mathbb{R}^3)$-$L^2(\mathbb{R}^3)$ is available for Euler-Maxwell equation…
Absence of Exponential Stability and Polynomial Stabilization in a Class of Beam Models with Tip Rotary Inertia
本文研究了一类梁模型中端部旋转惯性对稳定性的影响，发现混合耗散无法引起指数衰减，仅产生慢衰减行为。
We obtain some important fundamental inequalities concerning the long time behavior of high order derivatives for solutions of some dissipative systems in terms of their $L^2$ algebraic decay. Some of these inequalities have not been observed in the literature even for the fundamental Navier-Stokes equations.To illustrate this new approach, we derive bounds for the asymmetric incompressible fluids equations, where an improved inequalities established for the micro-rotational field. Finally, we show how this technique can be applied for other dissipative systems.
Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: application to the nonlinearly damped p-system
提出避开 Fourier 分析的物理空间双曲型亚强制性框架，在 SK/Kalman 秩条件下获得部分耗散双曲系统的精确及增强衰减估计，并应用于阻尼 Euler 方程与非线性阻尼 p-系统。
We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space $\mathbb{R}^d$ with $d\geq2$. Following our recent work [10] dedicated to the one-dimensional case, we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under consideration satisfies the so-called (SK) (for Shizuta-Kawashima) condition. Our results in particular apply to the compressible Euler system with damping in the velocity equation. Compared to the papers by Kawashima and Xu [27, 28] devoted to similar issu…
On decay of solutions to some perturbations of the Korteweg-de Vries equation
该研究聚焦加权及非对称加权Sobolev空间中Korteweg-de Vries方程耗散扰动解的正则性与衰减性关系，旨在揭示二者关联。
This work first gives the global existence and optimal decay rates of solutions to the classical Timoshenko system on the framework of Besov spaces. Due to the \textit{non-symmetric} dissipation, the general theory for dissipative hyperbolic systems ([30]) can not be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequen…
A note on asymptotics of linear dissipative kinetic equations in bounded domains
本文在有界域和一般边界条件下，利用能量估计、$L^2$-hypocoercivity与相对熵方法，建立了线性耗散动力学方程（含时间弛豫和Fokker-Planck模型）的$L^2$指数衰减性质，并导出了其在一般Maxwell边界条件下的扩散渐近性。
We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in the whole space, and study the asymptotic profile and optimal decay rates of solutions as time goes to infinity in L^{2}-sense. The operator L considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charao-Ikehata in 2020. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the…
Global decay of perturbations of equilibrium states for one-dimensional heat conducting compressible fluids of Korteweg type
本文研究一维Navier-Stokes-Fourier-Korteweg系统，通过引入考虑密度梯度的非线性扰动变量变换，证明其平衡态小扰动的全局存在性与渐近衰减。
Dissipative structure and decay rate for an inviscid non-equilibrium radiation hydrodynamics system
本文研究无粘非平衡辐射流体力学系统，证明多维局部解存在、一维模型具真正耦合性，提出非保守抛物平衡律熵函数并对称化一维系统，最终得到一维常平衡态扰动的全局存在性与时间衰减率。
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