From the Boltzmann equation to the Euler equations in the presence of boundaries

Golse, François

doi:10.1016/j.camwa.2012.02.009

Cited by 10 publications

(6 citation statements)

References 20 publications

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“…The fluid dynamic limits of boundary value problems for the Boltzmann equation are reviewed in [70] (see also [59] for a thorough discussion of the Stokes limit of the Boltzmann equation in the presence of boundaries). See also [11,31] for a discussion of the incompressible Euler limit, also in the presence of boundaries.…”

Section: Resultsmentioning

confidence: 99%

Fluid Dynamic Limits of the Kinetic Theory of Gases

Golse

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Section: Resultsmentioning

confidence: 99%

Fluid Dynamic Limits of the Kinetic Theory of Gases

Golse

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…It also motivates recent studies on time-dependent problems in its framework [10][11][12]. In the meantime, mathematical studies are also developing to include the argument of boundary condition for the fluid-dynamical equation [13][14][15].…”

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confidence: 87%

“…Proof of (16b) We use again the expression (13) of C[h] for small η and its consequence (15) for µζ < 0. We first rewrite (15) as…”

Section: B Sketch Of Derivation Of (16)mentioning

confidence: 99%

Second-Order Knudsen-Layer Analysis for the Generalized Slip-Flow Theory II: Curvature Effects

Hattori

Takata

2015

J Stat Phys

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Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists.

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“…This condition has been justified from the boundary condition at the microscopic scale in [15] for the Boltzmann equation. See also [44] or [66] for other examples of such derivations.…”

Section: Friction Boundary Conditionsmentioning

confidence: 99%

Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions

Coron¹,

Marbach²,

Sueur³

2016

Preprint

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In this work, we investigate the small-time global exact controllability of the Navier-Stokes equation, both towards the null equilibrium state and towards weak trajectories. We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time global exact controllability holds. Our analysis relies on the controllability of the Euler equation combined with asymptotic boundary layer expansions. Choosing the boundary controls with care enables us to guarantee good dissipation properties for the residual boundary layers, which can then be exactly canceled using local techniques.

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From the Boltzmann equation to the Euler equations in the presence of boundaries

Cited by 10 publications

References 20 publications

Fluid Dynamic Limits of the Kinetic Theory of Gases

Fluid Dynamic Limits of the Kinetic Theory of Gases

Second-Order Knudsen-Layer Analysis for the Generalized Slip-Flow Theory II: Curvature Effects

Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions

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