Sharpening of decay rates in Fourier based hypocoercivity methods

Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity

Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences diffusion at a rate proportional to $1/ν$, which is much faster than what would be produced by the static fluid if its viscosity is $0 < ν\ll 1$. This phenomenon is analyzed rigorously using the linear PDE governing the evolution of the solute. It is shown that the solution can be split into two pieces, an approximate solution and a remainder term. The approximate solution is governed by an infinite-dimensional syst…

Hypocoercivity without confinement

In this paper, hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement, in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analysis based on decoupled Fourier modes and a direct approach where, instead of the Poincaré inequality for the Dirichlet form, Nash's inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are …

无需 Fourier 分析的部分耗散双曲型系统大时间渐近：应用于非线性阻尼 p-系统

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。