The incompressible Navier–Stokes limit from the discrete-velocity BGK Boltzmann equation

Abstract In this paper, we extend the Bardos–Golse–Levermore program (Bardos et al 1993 Commun. Pure Appl. Math. 46 667–753) to prove that a local weak solution to the d -dimensional incompressible Navier–Stokes equations ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ) can be constructed by taking the hydrodynamic limit of a discrete-velocity Boltzmann equation with a simplified Bhatnagar–Gross–Krook collision operator. Moreover, in the case when the dimension is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , we characterise the combinations of finitely many particle velocities and probabilities that lead to the incompressible Navier–Stokes equations in the hydrodynamic limit. Numerical computations conducted in two-dimensional indicate that in the case of the simplest velocity lattice (D2Q9), the rate with which this hydrodynamic limit is achieved is of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> is the Knudsen number. For the future investigations, it is worth considering if the hydrodynamic limit of the discrete-velocity Boltzmann equation can be also rigorously justified in the presence of non-trivial boundary conditions.