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Ning Jiang - One of the best experts on this subject based on the ideXlab platform.
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From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions
arXiv: Analysis of PDEs, 2019Co-Authors: Ning Jiang, Yi-long LuoAbstract:For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigorously justified. This limit was justified in the recent breakthrough of Arsenio and Saint-Raymond \cite{Arsenio-SRM-2016} from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the corresponding scaling. In this sense, our result gives a classical solution analogue of the corresponding limit in \cite{Arsenio-SRM-2016}.
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boundary layers and incompressible navier stokes fourier limit of the boltzmann equation in bounded domain i
Communications on Pure and Applied Mathematics, 2017Co-Authors: Ning Jiang, Nader MasmoudiAbstract:We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions(-Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond [20,21] and Levermore and Masmoudi [28] to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond [34] for the linear Stokes-Fourier limit and Saint-Raymond [41] for the Navier-Stokes limit for hard potential kernels. Neither [34] nor [41] studied the damping of the acoustic waves. This paper extends the result of [34,41] to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai [46]. © 2016 Wiley Periodicals, Inc.
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Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann Equation in Bounded Domain (I)
arXiv: Analysis of PDEs, 2015Co-Authors: Ning Jiang, Nader MasmoudiAbstract:We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cut-off collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions-(Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to zero. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately, namely they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond \cite{Go-Sai04, Go-Sai05} and Levermore and Masmoudi \cite{LM} to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond \cite{M-S} for linear Stokes-Fourier limit and Saint-Raymond \cite{SRM} for Navier-Stokes limit for hard potential kernels. Both \cite{M-S} and \cite{SRM} didn't study the damping of the acoustic waves. This paper extends the result of \cite{M-S} and \cite{SRM} to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai \cite{Ukai}.
Nader Masmoudi - One of the best experts on this subject based on the ideXlab platform.
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boundary layers and incompressible navier stokes fourier limit of the boltzmann equation in bounded domain i
Communications on Pure and Applied Mathematics, 2017Co-Authors: Ning Jiang, Nader MasmoudiAbstract:We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions(-Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond [20,21] and Levermore and Masmoudi [28] to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond [34] for the linear Stokes-Fourier limit and Saint-Raymond [41] for the Navier-Stokes limit for hard potential kernels. Neither [34] nor [41] studied the damping of the acoustic waves. This paper extends the result of [34,41] to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai [46]. © 2016 Wiley Periodicals, Inc.
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Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann Equation in Bounded Domain (I)
arXiv: Analysis of PDEs, 2015Co-Authors: Ning Jiang, Nader MasmoudiAbstract:We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cut-off collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions-(Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to zero. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately, namely they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond \cite{Go-Sai04, Go-Sai05} and Levermore and Masmoudi \cite{LM} to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond \cite{M-S} for linear Stokes-Fourier limit and Saint-Raymond \cite{SRM} for Navier-Stokes limit for hard potential kernels. Both \cite{M-S} and \cite{SRM} didn't study the damping of the acoustic waves. This paper extends the result of \cite{M-S} and \cite{SRM} to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai \cite{Ukai}.
T. W. Körner - One of the best experts on this subject based on the ideXlab platform.
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Variations on a theme of Debs and Saint Raymond
Journal of the London Mathematical Society, 2008Co-Authors: T. W. KörnerAbstract:A famous theorem of Debs and Saint Raymond states that the complement of a set of first category is of strong multiplicity. We prove a theorem which combines this with a result of Rudin which states that independent closed sets of strong multiplicity exist. We also prove a theorem which combines the theorem of Debs and Saint Raymond with a theorem of Wiener and Wintner which states that there exists a measure with singular support whose convolution square is absolutely continuous
Yi-long Luo - One of the best experts on this subject based on the ideXlab platform.
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From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions
arXiv: Analysis of PDEs, 2019Co-Authors: Ning Jiang, Yi-long LuoAbstract:For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigorously justified. This limit was justified in the recent breakthrough of Arsenio and Saint-Raymond \cite{Arsenio-SRM-2016} from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the corresponding scaling. In this sense, our result gives a classical solution analogue of the corresponding limit in \cite{Arsenio-SRM-2016}.
Luo Yi-long - One of the best experts on this subject based on the ideXlab platform.
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From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions
2019Co-Authors: Jiang Ning, Luo Yi-longAbstract:For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigorously justified. This limit was justified in the recent breakthrough of Ars\'enio and Saint-Raymond \cite{Arsenio-SRM-2016} from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the corresponding scaling. In this sense, our result gives a classical solution analogue of the corresponding limit in \cite{Arsenio-SRM-2016}.Comment: 99 pages; all coments wellcome!. arXiv admin note: text overlap with arXiv:1604.01547 by other author
