Contents

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Convergence de la Solution d’une équation de Transport-diffusion vers la Solution d’une équation de Transport
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**Latifa Ait MAHIOUT ^{(1)} et Hisao Fujita Yashima^{(2)}
**

*Mathematics Subject Classification:* **(MSC 2010) 35K15, 35K58, 35L03, 35Q49.**

*Key words: *Transport-diffusion equation, transport equation, vanishing diffusion coefficient, approximate solutions

Abstract:

Transport equation and transport-diffusion equation obtained by
adding a diffusion term κΔu are considered in ℝ^{d}. For transport-diffusion
equation, we consider a family of approximate solutions which are defined by the
application of the integral operator corresponding to the fundamental solution of
the heat equation on each step of discretized time. For transport equation also we
consider a family of approximate solutions defined on the same family of time
discretizations. By using the estimate of the difference between approximate
solution for transport-diffusion equation and that for transport equation, we
prove the convergence of the solution of transport-diffusion equation to that of
transport equation when κ tends to 0. This result also asserts that the
difference between each term in the transport-diffusion equation and the
corresponding term in the transport equation is bounded proportionally to κ.