Exponential Stability Of The Quasigeostrophic Equation Under Random Perturbations

Existence of exponential attractor to a family of problems dominated by a perturbation of $p(x)$-laplacian with localized large diffusion via the $l$-trajectories method

This article is devoted to the study of the existence of an exponential attractor for a family of problems, in which diffusion $d_λ$ blows up in localized regions inside the domain, \begin{equation*} \begin{cases} \displaystyle u_t^λ-\mathrm{div}(d_λ(x)(|\nabla u^λ|^{p(x)-2}+η) \nabla u^λ)+ |u^λ|^{p(x)-2}u^λ=B(u^λ), & \mbox{ in } Ω\\ u^λ= 0, & \mbox{ on } \partialΩ\\ u^λ(0)=u^λ_0 \in L^2(Ω),& \end{cases} \end{equation*} and their limit problem via the $l$-trajectory method. \vskip .1 in \noindent {\it Mathematical Subject Classification 2010:} 35K55, 37L30, 35B40, 35B41. \newline {\it Key word…

泛进化方程最优控制中扰动的空间指数衰减

Spatial exponential decay of perturbations in optimal control of general evolution equations

本文研究了泛进化方程最优控制中空间局部扰动的指数衰减特性，证明了域均匀稳定性和可检测性对最优解的影响，并扩展到空间依赖传输速度和波动方程的情况。

Spectral Analysis and Long-Time Behaviour of a Fokker-Planck Equation with a Non-Local Perturbation

In this article we consider a Fokker-Planck equation with a non-local, mass preserving perturbation. We show that the perturbed Fokker-Planck operator generates a $C_0$-semigroup on an exponentially weighted $L^2$-space. Surprisingly, the spectrum of the Fokker-Planck operator is not affected by the perturbation. In particular there still exists a unique (normalized) stationary solution of the perturbed equation. And we have convergence towards the stationary state with exponential rate -1, the same rate as for the unperturbed Fokker-Planck equation. Moreover, for any $k\in\mathbb N$ there exi…

半群生成元的相对有界扰动的均匀稳定性

Uniform stabilization for relatively bounded perturbations of generators of semigroup

研究线性算子生成半群在相对有界扰动下指数稳定性的鲁棒性，推广Gibson稳定性定理，并应用于均匀抛物方程、退化/奇异抛物方程及耦合板系统。

Asymptotic Stability for KdV Solitons in Weighted $H^s$ Spaces

In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.

Stability of a planar front in a multidimensional reaction-diffusion system

We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A-priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that…

Asymptotic stability of traveling wave solutions for perturbations with algebraic decay

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of thi…

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg-de Vries-Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

Exponential approximations for the primitive equations of the ocean

We show that in the limit of small Rossby number $\eps$, the primitive equations of the ocean (OPEs) can be approximated by ``higher-order quasi-geostrophic equations'' up to an exponential accuracy in $\eps$. This approximation assumes well-prepared initial data and is valid for a timescale of order one (independent of $\eps$). Our construction uses Gevrey regularity of the OPEs and a classical method to bound errors in higher-order perturbation theory.

Exponential decay for damped Klein-Gordon equations on asymptotically cylindrical and conic manifolds

We study the decay of the global energy for the damped Klein-Gordon equation on non-compact manifolds with finitely many cylindrical and subconic ends up to bounded perturbation. We prove that under the Geometric Control Condition, the decay is exponential, and that under the weaker Network Control Condition, the decay is logarithmic, by developing the global Carleman estimate with multiple weights.

Cutoff Boltzmann equation with polynomial perturbation near Maxwellian

In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $γ\in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, w…

边界值问题一般一维双曲系统的指数分裂的鲁棒性

Robustness of Exponential Dichotomies of Boundary-Value Problems for General First-Order Hyperbolic Systems

本文研究了一般一维双曲系统边界值问题指数分裂在小扰动下的鲁棒性，证明其在连续函数空间中保持不变。