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Diffusion approximation of radiative transfer problems with interfaces. (English) Zbl 0976.45008
The authors derive the diffusion approximation of transport equations with discontinuities at interfaces between different media. More exactly, they assume that the geometry and the physical parameters are invariant by translations along directions orthogonal to $$e_3=(0,0,1)$$. After rescaling, the problem to be dealt with is the following: ${\mu \over \varepsilon}D_x a^i_\varepsilon(x,\mu) + {1 \over \varepsilon^2} \int_{-1}^1 \sigma^i(\mu,\mu')[a^i_\varepsilon(x,\mu)-a^i_\varepsilon(x,\mu')]d\mu' + \Sigma_a a^i_\varepsilon(x,\mu) = S(x),$ $(x,\mu)\in X^i\times [-1,1],$ $a^i_\varepsilon(\mu) = \sum_{j=1}^2 R^{i,j}(a^i_\varepsilon)(\mu),\quad \text{on } x=0,\;\mu \in \Gamma_-^i,\;i=1,2,$ where $$\varepsilon >0$$, $$X^1=\mathbb{R}_+$$, $$X^2=\mathbb{R}_-$$, $$\Gamma_-^1=[0,1]$$, $$\Gamma_-^2=[-1,0]$$ and $$\Sigma_a$$ denotes some absorption coefficient. Moreover, the right-hand side $$S$$ is assumed to be smooth and compactly supported, while the kernel $$\sigma^i$$ is assumed to be bounded and strictly positive with $$\Sigma^i=\int_{-1}^1 \sigma^i(\mu,\mu')d\mu$$ being independent of $$\mu'\in \Gamma_-^i$$, $$i=1,2$$.
The authors show that, under suitable assumptions on the reflection operators, $$R^{11}$$ and $$R^{22}$$, and the transmission ones, $$R^{12}$$ and $$R^{21}$$, the family $$(a^1_\varepsilon,a^2_\varepsilon)$$ converges, as $$\varepsilon \to 0+$$, strongly in $$L^1(X^1\times [-1,1])\times L^1(X^2\times [-1,1])$$ to a pair $$(u^1,u^2)\in H^1(X^1)\times H^1(X^2)$$ satisfying an explicit parabolic transmission problem.
Finally, since the diffusion approximation is not correct in a neighbourhood of the interface, the authors also prove that the transport solution can be approximated by a diffusion vector-term plus an interface vector-layer decaying exponentially fast, away from the interface.
Some examples of admissible interface conditions are given in the last section of the paper.

##### MSC:
 45K05 Integro-partial differential equations 74J10 Bulk waves in solid mechanics 85A25 Radiative transfer in astronomy and astrophysics
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