On the Strong Solution of the Ghost Effect System

In this paper, the ghost effect system modeling the ghost effect phenomena for rarefied gas is investigated. It is shown that any strong solution of the Cauchy problem moves along the direction from the low temperature part to the high part at least on a nonzero measure domain for any time. This also holds for the initial boundary value problem with additional boundary conditions. The global well-posedness of the strong solution is further obtained for the 2 dimensional ghost effect system under some conditions on the temperature, while the initial velocity could be arbitrarily large.

[1]  Claude Bardos,et al.  The Classical Incompressible Navier-Stokes Limit of the Boltzmann Equation(Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics) , 1991 .

[2]  Raffaele Esposito,et al.  Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation , 1989 .

[3]  Yoshio Sone,et al.  Molecular gas dynamics , 2007 .

[4]  Yoshio Sone,et al.  Molecular Gas Dynamics: Theory, Techniques, and Applications , 2006 .

[5]  I. I. Bakelʹman,et al.  Geometric Analysis and Nonlinear Partial Differential Equations , 1993 .

[6]  Konstantina Trivisa,et al.  A Low Mach Number Limit of a Dispersive Navier-Stokes System , 2012, SIAM J. Math. Anal..

[7]  Nader Masmoudi,et al.  From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System , 2010 .

[8]  Shi-Hsien Yu,et al.  Hydrodynamic limits with shock waves of the Boltzmann equation , 2005 .

[9]  Y. Sone.  Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit , 2000 .

[10]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[11]  Laure Saint-Raymond,et al.  The Incompressible Navier-Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials , 2008, 0808.0039.

[12]  Ping Zhang,et al.  On the global well-posedness of 2-D inhomogeneous incompressible Navier–Stokes system with variable viscous coefficient , 2015 .

[13]  Kiyoshi Asano,et al.  The Euler limit and initial layer of the nonlinear Boltzmann equation , 1983 .

[14]  Feimin Huang,et al.  Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities , 2009, 0904.1836.

[15]  Yi Wang,et al.  Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity , 2010, 1011.1990.

[16]  F. Golse,et al.  Fluid dynamic limits of kinetic equations. I. Formal derivations , 1991 .

[17]  François Golse,et al.  Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation , 1993 .

[18]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[19]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[20]  J. Maxwell,et al.  On Stresses in Rarified Gases Arising from Inequalities of Temperature , 2022 .

[21]  Claude Bardos,et al.  KINETIC EQUATIONS: FLUID DYNAMICAL LIMITS AND VISCOUS HEATING , 2008 .

[22]  F. Huang.  Thermal creep flow for the Boltzmann equation , 2015 .

[23]  Yan Guo,et al.  Boltzmann diffusive limit beyond the Navier‐Stokes approximation , 2006 .

[24]  LOCAL WELL-POSEDNESS OF A GHOST EFFECT SYSTEM , 2009 .

[25]  Pierre-Louis Lions,et al.  From the Boltzmann Equations¶to the Equations of¶Incompressible Fluid Mechanics, I , 2001 .

[26]  Tai-Ping Liu,et al.  Thermal transpiration for the linearized Boltzmann equation , 2007 .

[27]  Tong Yang,et al.  Justification of Diffusion limit for the Boltzmann Equation with a non-trivial Profile , 2015, 1505.02258.

[28]  S. Ukai.  The incompressible limit and the initial layer of the compressible Euler equation , 1986 .

[29]  Zhouping Xin,et al.  Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier–Stokes equations , 2010 .

[30]  Yong Wang,et al.  The Limit of the Boltzmann Equation to the Euler Equations for Riemann Problems , 2011, SIAM J. Math. Anal..

[31]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

[32]  François Golse,et al.  The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels , 2004 .