Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.Comment: 17 pages, 1 figur

Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations

textabstractThermal bath coupling mechanisms as utilized in molecular dynamics are applied to partial differential equation models. Working from a semi-discrete (Fourier mode) formulation for the Burgers–Hopf or Korteweg–de Vries equation, we introduce auxiliary variables and stochastic perturbations in order to drive the system to sample a target ensemble which may be a Gibbs state or, more generally, any smooth distribution defined on a constraint manifold. We examine the ergodicity of approaches based on coupling of the heat bath to the high wave numbers, with the goal of controlling the ensemble through the fast modes. We also examine different thermostat methods in the extent to which dynamical properties are corrupted in order to accurately compute the average of a desired observable with respect to the invariant distribution. The principal observation of this paper is that convergence to the invariant distribution can be achieved by thermostatting just the highest wave number, while the evolution of the slowest modes is little affected by such a thermostat

Regularisation and Long-Time Behaviour of Random Systems

Schenke A. Regularisation and Long-Time Behaviour of Random Systems. Bielefeld: Universität Bielefeld; 2020.In this work, we study several different aspects of systems modelled by partial differential equations (PDEs), both deterministic and stochastically perturbed. The thesis is structured as follows: Chapter I gives a summary of the contents of this work and illustrates the main results and ideas of the rest of the thesis. Chapter II is devoted to a new model for the flow of an electrically conducting fluid through a porous medium, the tamed magnetohydrodynamics (TMHD) equations. After a survey of regularisation schemes of fluid dynamical equations, we give a physical motivation for our system. We then proceed to prove existence and uniqueness of a strong solution to the TMHD equations, prove that smooth data lead to smooth solutions and finally show that if the onset of the effect of the taming term is deferred indefinitely, the solutions to the tamed equations converge to a weak solution of the MHD equations. In Chapter III we investigate a stochastically perturbed tamed MHD (STMHD) equation as a model for turbulent flows of electrically conducting fluids through porous media. We consider both the problem posed on the full space $\R^{3}$ as well as the problem with periodic boundary conditions. We prove existence of a unique strong solution to these equations as well as the Feller property for the associated semigroup. In the case of periodic boundary conditions, we also prove existence of an invariant measure for the semigroup. The last chapter deals with the long-time behaviour of solutions to SPDEs with locally monotone coefficients with additive L\'{e}vy noise. Under quite general assumptions, we prove existence of a random dynamical system as well as a random attractor. This serves as a unifying framework for a large class of examples, including stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$\alpha$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard-type equations, stochastic Kuramoto-Sivashinsky-type equations, stochastic porous media equations and stochastic $p$-Laplace equations