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On Some Models in Radiation Hydrodynamics

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Research in Mathematics of Materials Science

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 31))

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Abstract

The paper is a review on the problem of the compressible radiation hydrodynamics. We focus on the weak solutions of the full viscous system coupled with radiation and their generalization (semi-relativistic case) in a bounded domain \(\varOmega \subset \mathbb {R}^3\). Moreover, we study the strong solutions of the inviscid system with damping term or Euler–Maxwell’s system coupled with radiation in the whole space \(\mathbb {R}^3\).

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Notes

  1. 1.

    Note that (2.22) implicitly includes the initial condition

    $$\displaystyle \begin{aligned} \varrho(0, \cdot) = \varrho_0.\end{aligned} $$
  2. 2.

    As the viscous stress contains first derivatives of the velocity u, for (2.23) to make sense, the field u must belong to a certain Sobolev space with respect to the spatial variable. Here, we require that \( {\mathbf u} \in L^2(0,T; W^{1,2}_0 (\varOmega )) \).

  3. 3.

    Let us recall [1] the formula for the entropy of a photon gas

    $$\displaystyle \begin{aligned} s^R=-\frac{2k}{c^3}\int_0^{\infty}\int_{{\mathcal S}^2} \nu^2\left[ n\log n-(n+1)\log(n+1)\right]d{\mathbf \omega} d\nu, \end{aligned} $$
    (2.43)

    where \(n=n(I)=\frac {c^2I}{2h\alpha ^3\nu ^3}\) is the occupation number. Defining the radiative entropy flux

    $$\displaystyle \begin{aligned} {\mathbf q}^R=-\frac{2k}{c^2}\int_0^{\infty}\int_{{\mathcal S}^2} \nu^2\left[ n\log n-(n+1)\log(n+1)\right]{\mathbf \omega}\ d{\mathbf \omega} d\nu, \end{aligned} $$
    (2.44)

    and using the radiative transfer equation, we get the equation

    $$\displaystyle \begin{aligned} \partial_t s^R+\mbox{div}_x {\mathbf q}^R =-\frac{k}{h}\int_0^{\infty}\int_{{\mathcal S}^2} \frac{1}{\nu} \log \frac{n}{n+1}\ S\ d{\mathbf \omega} d\nu=:\varsigma^R. \end{aligned} $$
    (2.45)

    Moreover, \(\log \frac {n(B)}{n(B)+1}=-\frac {h\nu }{k \vartheta }\Big (1-\alpha \frac {\mathbf \omega \cdot \mathbf u}{c}\Big )\) and

    $$\displaystyle \begin{aligned} \alpha=\frac{\sigma_a+\sigma_s}{\sigma_a+2\sigma_s}, \end{aligned} $$
    (2.46)

    For more details, see [18].

  4. 4.
    $$\displaystyle \begin{aligned} A_1:= \left( \begin{array}{cccccc} 0 & \overline{\varrho} & 0 & 0 & 0 & 0\\\ \alpha & 0 & 0 & 0 & \beta & \frac{1}{3\overline{\varrho}}\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & \gamma & 0 & 0 & 0 & 0\\\ 0 & \delta & 0 & 0 & 0 & 0 \end{array} \right) ,\ \ A_2:= \left( \begin{array}{cccccc} 0 & 0 & \overline{\varrho} & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0\\\ \alpha & 0 & 0 & 0 & \beta & \frac{1}{3\overline{\varrho}}\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & \gamma & 0 & 0 & 0 \\\ 0 & 0 & \delta & 0 & 0 & 0 \end{array} \right) ,\ \end{aligned}$$
    $$\displaystyle \begin{aligned} A_3:= \left( \begin{array}{cccccc} 0 & 0 & 0 & \overline{\varrho} & 0 & 0\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ \alpha & 0 & 0 & 0 & \beta & \frac{1}{3\overline{\varrho}}\\\ 0 & 0 & 0 & \gamma & 0 & 0\\\ 0 & 0 & 0 & \delta & 0 & 0 \end{array} \right), \end{aligned}$$

    and

    $$\displaystyle \begin{aligned} D:= \left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0 & \mu & 0\\\ 0 & 0 & 0 & 0 & 0 & \nu \end{array} \right) ,\ \ B:= \left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0\\\ 0 & \nu & 0 & 0 & 0 & 0\\\ 0 & 0 & \nu & 0 & 0 & 0\\\ 0 & 0 & 0 & \nu & 0 & 0\\\ 0 & 0 & 0 & 0 & \zeta & -\eta\\\ 0 & 0 & 0 & 0 & -\pi & \sigma_a \end{array} \right), \end{aligned}$$
  5. 5.
    $$\displaystyle \begin{aligned} \alpha= \frac{\overline{p}_{\varrho}}{\overline{\varrho}},\quad \beta=\frac{\overline{p}_{\vartheta}}{\overline{\varrho}},\quad \gamma=\frac{\overline{\vartheta}\overline{p}_{\vartheta}}{\overline{\varrho}\overline{C}_v},\quad \delta=\frac{4}{3}\overline{E}_r,\quad \mu= \frac{\kappa}{\overline{\varrho}\overline{C}_v},\end{aligned}$$
    $$\displaystyle \begin{aligned} \tau=\frac{1}{3\sigma_s},\quad \zeta= \frac{4a\sigma_a\overline{\vartheta}^3}{\overline{\varrho}\overline{C}_v},\quad \eta=\frac{\sigma_a}{\overline{\varrho}\overline{C}_v},\quad \pi=4a\sigma_a\overline{\vartheta}^3. \end{aligned}$$

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Acknowledgements

The work of Š.N. were supported by the Czech Science Foundation grant GA19-04243S and by the programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan 67985840.

Author information

Authors and Affiliations

  1. Université Paris Cité and Sorbonne Université, CNRS, Paris, France

    Xavier Blanc

  2. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, Créteil, France

    Bernard Ducomet

  3. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic

    Šárka Nečasová

Authors
  1. Xavier Blanc
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  2. Bernard Ducomet
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Correspondence to Šárka Nečasová .

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Editors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA

    Malena I. Español

  2. Mathematics Department, University of Pittsburgh, Pittsburgh, PA, USA

    Marta Lewicka

  3. Department of Mathematics, Heriot-Watt University, Edinburgh, UK

    Lucia Scardia

  4. Institute of Mathematics, Universität Würzburg, Würzburg, Germany

    Anja Schlömerkemper

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Blanc, X., Ducomet, B., Nečasová, Š. (2022). On Some Models in Radiation Hydrodynamics. In: Español, M.I., Lewicka, M., Scardia, L., Schlömerkemper, A. (eds) Research in Mathematics of Materials Science. Association for Women in Mathematics Series, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-031-04496-0_4

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