Self-adjoint Laplacians and symmetric diffusions on hyperbolic attractors

Self-adjoint Laplacians on partially and generalized hyperbolic attractors

We construct self-adjoint Laplacians and symmetric Markov semigroups on partially hyperbolic attractors and on hyperbolic attractors with singularities, endowed with Gibbs u-measures. If the measure has full support, we can also guarantee the existence of an associated symmetric Hunt diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on manifolds of negative sectional curvature the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians.

具有分数阶拉普拉斯算子的非线性双曲-抛物耦合系统吸引子的上半连续性

Upper semicontinuity of the attractor for a nonlinear hyperbolic-parabolic coupled system with fractional Laplacian

本文研究了具有分数阶拉普拉斯算子的二维非线性双曲-抛物耦合系统，证明了全局解的存在唯一性，以及半群吸引子的存在性、正则性、上半连续性和有限分形维数指数吸引子的存在性。

Forward and Adjoint Sensitivity Computation of Chaotic Dynamical Systems

This paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems, such as the Lorenz attractor. The algorithms compute the derivative of long time averaged "statistical" quantities to infinitesimal perturbations of the system parameters. The algorithms are demonstrated on the Lorenz attractor. We show that sensitivity derivatives of statistical quantities can be accurately estimated using a single, short trajectory (over a time interval of 20) on the Lorenz attractor.

Isospectrum of non-self-adjoint almost-periodic Schrodinger operators

For non-self-adjoint almost-periodic Schrödinger operators, a criterion is given to guarantee that they have both the same spectrum and same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the Moser-Pöschel argument for opening gaps may not be valid for non-self-adjoint operators.

Attractors for Damped Semilinear Wave Equations with a Robin--Acoustic Boundary Perturbation

Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-Δu + u + f(u)=0 \] on a bounded domain $Ω$ in $\mathbb{R}^3$ with a perturbation parameter $\varepsilon>0$ occurring in an acoustic boundary condition, limiting ($\varepsilon=0$) to a Robin boundary condition. With minimal assumptions on the nonlinear term $f$, the existence and uniqueness of global weak solutions is shown for each $\varepsilon\in[0,1]$. Also, the existence of a family of global attractors is shown to exist. After proving a general result concerning the upper-semicontinuity of a one-parameter family of se…

A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations

We consider the semilinear reaction diffusion equation $\partial_tϕ-νΔϕ-V(x)ϕ+f(ϕ)=0$, $ν>0$ in a bounded domain $Ω\subset\mathbb{R}^N$. We assume the standard Allen-Cahn-type nonlinearity, while the potential $V$ is either the inverse square potential $V(x)=δ|x|^{-2}$ or the borderline potential $V(x)=δ\mathrm{dist}(x,\partialΩ)^{-2}$, $δ\geq 0$ (thus including the classical Allen-Cahn equation as a special case when $δ=0$). In the subcritical cases $δ=0$, $N\geq 1$ and $0<μ:=\fracδν<μ^*$, $N\geq 3$ (where $μ^*$ is the optimal constant of Hardy and Hardy-type inequalities), we present a new e…