Energy estimates and hypocoercivity analysis for a multi-phase Navier-Stokes-Vlasov-Fokker-Planck system with uncertainty

TL;DR

This paper analyzes a coupled Navier-Stokes-Vlasov-Fokker-Planck system with uncertainty, establishing energy decay and regularity results, and demonstrating spectral accuracy of a stochastic Galerkin method for long-term simulations.

Contribution

It provides the first uniform regularity and exponential decay results for the uncertain multi-phase kinetic-fluid system, and proves spectral accuracy of the gPC-sG method in this context.

Findings

Energy decays exponentially over time.

Uniform regularity for random initial data is established.

gPC-sG method achieves spectral accuracy and exponential error decay.

Abstract

This paper is concerned with a kineitc-fluid model with random initial inputs in the fine particle regime, which is a system coupling the incompressible Navier-Stokes equations and the Vlasov-Fokker-Planck equations that model dispersed particles of different sizes. A uniform regularity for random initial data near the global equilibrium is established in some suitable Sobolev spaces by using energy estimates, and we also prove the energy decays exponentially in time by hypocoercivity arguments, which means that the long time behavior of the solution is insensitive to the random perturbation in the initial data. For the generalized polynomial chaos stochastic Galerkin method (gPC-sG) for the model, with initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Knudsen number, and the error decays exponentially in time.