Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations

Abstract

We present a rigorous convergence result for the smooth solutions to a singular semilinear hyperbolic approximation, a vector BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof is based on the use of a constant right symmetrizer, weighted with respect to the parameter of the singular pertubation system. This symmetrizer provides a conservative-dissipative form for the system and this allow us to perform uniform energy estimates and to get the convergence by compactness.

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

“…Thus, the estimates in [6] still hold here: the quadratic pressure only provides an additional quadratic term in the fifth and the ninth line of the nonlinear vector N (w +w) in [ [6], (26)]. These supplementary quadratic terms can be handled exactly as the other ones in the energy estimates in [6].…”

“…Preliminary results. Here we collect some preliminary results, which hold for local times, essentially due to our previous work [6]. Let us start with the following remark.…”

“…Of course T ε could depend on ε. In the following, we recall and adapt some results from [6], showing that there exist ε 0 and a fixed and positive time T * , independent of ε and depending on the Sobolev norm of the initial data, such that, for ε ≤ ε 0 , some local in time H s -estimates on the solutions to the approximating system hold uniformly with respect to ε. In this context, we consider the constant vector (ρ, 0, 0) and the translated variables…”

“…We prove the strong convergence in the Sobolev spaces, for any interval of time [0, T ], T > 0, of the vector-BGK model presented in (1.3) to the incompressible Navier-Stokes equations on the two-dimensioanl torus. To achieve this result, the novelty relies in using local in time H s -estimates from a previous work, see [6], combined with the L 2 -relative entropy estimate and the standard interpolation Theorem. More precisely, part of the results of [6] provides uniform (in ε) estimates of Gronwall type in the Sobolev spaces, which hold in [0, T * ], where T * > 0 is depending on a fixed constant M > 0 and on the norm of the initial data.…”

“…In [[6], (18)-(19)], we consider a translated version of the relaxation system (2.6). Of course this is an equivalent formulation of the approximating model, and since the translation vector (ρ, 0, 0) in [[6],(18)] is constant in t and x, most of the energy estimates in[6] can be used here.• A further change of variables, involving the dissipative constant right symmetrizer Σ in [[6],(28)] is defined in [[6],(30)]. However, here the energy estimates from[6] are expressed in terms of the original relaxation variables (2.5) to avoid further complications.…”