Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side
Applications of Mathematics, Tome 55 (2010) no. 2, pp. 111-149
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We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the $L^p$ norm of the heat sources for exponents $p$ close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating), with many applications such as crystal growth.
We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the $L^p$ norm of the heat sources for exponents $p$ close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating), with many applications such as crystal growth.
DOI :
10.1007/s10492-010-0005-9
Classification :
35D05, 35D30, 35K05, 35K15, 35K20, 35K55, 35K59, 80A20
Keywords: radiative heat transfer; nonlinear parabolic equation; nonlocal boundary condition; right-hand side in $L^1$
Keywords: radiative heat transfer; nonlinear parabolic equation; nonlocal boundary condition; right-hand side in $L^1$
@article{10_1007_s10492_010_0005_9,
author = {Druet, Pierre-Etienne},
title = {Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side},
journal = {Applications of Mathematics},
pages = {111--149},
year = {2010},
volume = {55},
number = {2},
doi = {10.1007/s10492-010-0005-9},
mrnumber = {2600939},
zbl = {1224.35057},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0005-9/}
}
TY - JOUR AU - Druet, Pierre-Etienne TI - Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side JO - Applications of Mathematics PY - 2010 SP - 111 EP - 149 VL - 55 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0005-9/ DO - 10.1007/s10492-010-0005-9 LA - en ID - 10_1007_s10492_010_0005_9 ER -
%0 Journal Article %A Druet, Pierre-Etienne %T Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side %J Applications of Mathematics %D 2010 %P 111-149 %V 55 %N 2 %U http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0005-9/ %R 10.1007/s10492-010-0005-9 %G en %F 10_1007_s10492_010_0005_9
Druet, Pierre-Etienne. Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side. Applications of Mathematics, Tome 55 (2010) no. 2, pp. 111-149. doi: 10.1007/s10492-010-0005-9
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