Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Abstract

The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time [0, T ], with T > 0. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time H s -estimates for the model, established in a previous work, combined with the L 2 -relative entropy estimate and the interpolation properties of the Sobolev spaces.

The relaxation limit of bipolar fluid models

“…Regarding the structure of the momenum equations of system (2), observe that the frictional coefficient 1/ε also multiplies the internal energy and electric field terms. From the bipolar Boltzmann-Poisson model with a Lenard-Bernstein collision operator [10,17,19], one formally derives the following system (see appendix for details)…”

“…Using the symmetry of N, one derives the formulas (5) for the functional derivatives δE δρ , δE δn . Introducing (5) to (2) leads to (11), where p 1 , p 2 are the pressures connected to the internal energies via the usual thermodynamic formulas (6). The formal relaxation limit of (2) is the system (1); establishing this limit is the objective of the present work.…”

“…This approach was successful for the relaxation limit in single-species fluid models [15,3], as well as for certain (weakly coupled through friction) multicomponent systems [9]. The relative energy method provides an efficient mathematical mechanism for stability analysis and establishing limiting processes; see [5] for early developments, [2,15,16] and references therein for applications to diffusive relaxation. Here, the bipolar fluid models are considered in a bounded domain, which is closer to the actual physical situation and requires handling the boundary conditions.…”

“…Here, the bipolar fluid models are considered in a bounded domain, which is closer to the actual physical situation and requires handling the boundary conditions. No-flux boundary conditions are applied to the velocities for (2), to the fluxes for (1), and to the electric field for both systems.…”

Weak-Strong Uniqueness and High-Friction Limit for Euler-Riesz Systems

“…Since its origin, e.g. [14], this method has seen an extensive applicability to diffusive relaxation [2,3,9,17,18].…”