Global Regularity and Multiscale Approach for Thermal Radiation Heat Transfer

This paper discusses the multiscale analysis for the thermal radiation heat transfer in composite materials with a periodic microstructure. The new contributions reported in this paper are threefold: global existence of the solution for the nonlinear equation with discontinuous coefficients is proved by using Rothe's method, the multiscale asymptotic expansions of the solution for the thermal radiation heat transfer equation with rapidly oscillating coefficients are presented, and the convergence results between the original solution $u^\varepsilon(x,t)$ and the multiscale asymptotic solutions with an explicit convergence rate are obtained.

[1]  J. Cui,et al.  Interior Hölder and gradient estimates for the homogenization of the linear elliptic equations , 2013 .

[2]  Louis Nirenberg,et al.  Estimates for elliptic systems from composite material , 2003 .

[3]  Ya Zhang,et al.  Multiscale Asymptotic Method for Maxwell's Equations in Composite Materials , 2010, SIAM J. Numer. Anal..

[4]  P. Grisvard.  Boundary value problems in non-smooth domains , 1980 .

[5]  Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems , 2010 .

[6]  Pierre Charrier,et al.  Asymptotic Transport Models for Heat and Mass Transfer in Reactive Porous Media , 2003, Multiscale Model. Simul..

[7]  H. Amann.  Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[8]  V. V. Zhikov,et al.  AVERAGING OF PARABOLIC OPERATORS WITH ALMOST PERIODIC COEFFICIENTS , 1983 .

[9]  T. Hou,et al.  Multiscale Finite Element Methods for Nonlinear Problems and Their Applications , 2004 .

[10]  M. Avellaneda,et al.  Compactness methods in the theory of homogenization , 1987 .

[11]  Walter Allegretto,et al.  Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients , 2007 .

[12]  R. Bruce Lindsay,et al.  Nonhomogeneous Media and Vibration Theory, by Enrique Sanchez‐Palencia , 1981 .

[13]  Walter W. Yuen,et al.  COMBINED CONDUCTIVE/RADIATIVE HEAT TRANSFER IN HIGH POROSITY FIBROUS INSULATION MATERIALS: THEORY AND EXPERIMENT , 2003 .

[14]  G. M. Lieberman.  SECOND ORDER PARABOLIC DIFFERENTIAL EQUATIONS , 1996 .

[15]  M. Vogelius,et al.  Gradient Estimates for Solutions to Divergence Form Elliptic Equations with Discontinuous Coefficients , 2000 .

[16]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[17]  Grégoire Allaire,et al.  Homogenization of a Conductive and Radiative Heat Transfer Problem , 2008, Multiscale Model. Simul..

[18]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[19]  N. Svanstedt.  G—convergence of parabolic operators , 1999 .

[20]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[21]  O. Oleinik,et al.  Mathematical Problems in Elasticity and Homogenization , 2012 .

[22]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[23]  M. Pinar Mengüç,et al.  Thermal Radiation Heat Transfer , 2020 .

[24]  G. Burton.  Sobolev Spaces , 2013 .

[25]  P. Donato,et al.  An introduction to homogenization , 2000 .

[26]  H. Amann.  Non-local quasi-linear parabolic equations , 2005 .

[27]  Yalchin Efendiev,et al.  Numerical Homogenization of Nonlinear Random Parabolic Operators , 2004, Multiscale Model. Simul..

[28]  Alexander Pankov,et al.  G-Convergence and Homogenization of Nonlinear Partial Differential Operators , 1997 .