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Compressible Euler limit from Boltzmann equation with Maxwell reflection boundary condition in half-space
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abstract
Starting from the local-in-time classical solution to the compressible Euler system with impermeable boundary condition in half-space, by employing the coupled weak viscous layers (governed by linearized compressible Prandtl equations with Robin boundary condition) and linear kinetic boundary layers, and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some new boundary estimates both for Prandtl and Knudsen layers, we proved the local-in-time existence of Hilbert expansion type classical solutions to the scaled Boltzmann equation with Maxwell reflection boundary condition with accommodation coefficient $\alpha_\varepsilon=O(\sqrt{\varepsilon})$ when the Knudsen number $\varepsilon$ small enough. As a consequence, this justifies the corresponding case of formal analysis in Sone's books \cite{Sone-2002book, Sone-2007-Book}. This also extends the results in \cite{GHW-2020} from specular to Maxwell reflection boundary condition. Both of this paper and \cite{GHW-2020} can be viewed as generalizations of Caflisch's classic work \cite{Caflish-1980-CPAM} to the cases with boundary.
Forward citations
Cited by 4 Pith papers
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Kinetic-fluid boundary layers and acoustic limit for the Boltzmann equation with general Maxwell reflection boundary condition
The acoustic limit from the hard-sphere Boltzmann equation with Maxwell reflection boundary condition is proved for classical solutions for all accommodation coefficients 0<α≤1, with convergence rate ε^{1/4}.
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Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits
For hard-sphere Boltzmann with Maxwell reflection and 0<α<1, the nonlinear Knudsen layer equation is well-posed in weighted L∞ and its far-field state is determined by the source terms.
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Knudsen boundary layer equations with incoming boundary condition: full range of cutoff collision kernels and Mach numbers of the far field
Existence, uniqueness, and exponential decay for Knudsen layer solutions with incoming data are established for the full range of cutoff kernels and all Mach numbers.
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