Low Mach number asymptotics for reacting compressible fluid flows (Q2268679)

For each \( \varepsilon>0\), it is assumed the existence of a solution \((\rho_\varepsilon,{\mathbf u}_\varepsilon)\) in the sense of renormalized solution, \(\vartheta_\varepsilon\) in the sense of very weak solution and \(Y_\varepsilon\) in the sense of entropy solution to the Navier-Stokes-Fourier system for classical perfect gases, under homogeneous boundary conditions, \[ \partial_t\rho_\varepsilon+\nabla\cdot(\rho_\varepsilon{\mathbf u}_\varepsilon)=0; \] \[ \begin{multlined}\partial_t(\rho_\varepsilon{\mathbf u}_\varepsilon)+\nabla\cdot(\rho_\varepsilon{\mathbf u}_\varepsilon \otimes {\mathbf u}_\varepsilon)- \nabla\cdot \left(2\mu(\vartheta_\varepsilon)D{\mathbf u}_\varepsilon -{2\over 3}\mu(\vartheta_\varepsilon) \nabla\cdot {\mathbf u}_\varepsilon I\right)\\ =-{1\over\varepsilon^2} \left(a\vartheta_\varepsilon^3\nabla \vartheta_\varepsilon +\nabla( {\vartheta_\varepsilon^{5/2}}P\left({\rho_\varepsilon\over\vartheta_\varepsilon ^{3/2}}\right))\right)- {1\over\varepsilon}\rho_\varepsilon{\mathbf j};\end{multlined} \] \[ \begin{multlined}\partial_t(\rho_\varepsilon s)+\nabla\cdot(\rho_\varepsilon s{\mathbf u}_\varepsilon) -\nabla\cdot\left({\kappa(\vartheta_\varepsilon)\over\vartheta_\varepsilon}\nabla\vartheta_\varepsilon \right)=\varepsilon^2 {2\mu(\vartheta_\varepsilon)\over\vartheta_\varepsilon} \left(|D{\mathbf u}_\varepsilon|^2 -{1\over 3}(\nabla\cdot {\mathbf u}_\varepsilon)^2\right)\\ +{\kappa(\vartheta_\varepsilon)\over\vartheta_\varepsilon^2}|\nabla\vartheta_\varepsilon|^2 +{h\over\varepsilon}\rho_\varepsilon Y_\varepsilon {(\vartheta_\varepsilon-\bar\vartheta)^+\over\vartheta_\varepsilon}\exp\left(-{k\over \vartheta_\varepsilon-\bar\vartheta}\right); \end{multlined} \] \[ \partial_t(\rho_\varepsilon Y_\varepsilon)+\nabla\cdot(\rho_\varepsilon Y_\varepsilon {\mathbf u}_\varepsilon)-\nabla\cdot(d(\vartheta_\varepsilon)\nabla Y_\varepsilon)= -{1\over\varepsilon}\rho_\varepsilon Y _\varepsilon (\vartheta_\varepsilon-\bar\vartheta)^+\exp\left(-{k\over \vartheta_\varepsilon-\bar\vartheta}\right); \] \[ {d\over dt}\int_\Omega \rho_\varepsilon\left( \varepsilon^2{|{\mathbf u}_\varepsilon |^2\over 2}+a{\vartheta_\varepsilon^4\over\rho_\varepsilon}+ {3\over 2} {\vartheta_\varepsilon^{5/2}\over\rho_\varepsilon}P\left({\rho_\varepsilon \over\vartheta_\varepsilon^{3/2}}\right)+ hY_\varepsilon+ \varepsilon x_3\right)dx=0. \] Here \(\rho_\varepsilon\), \({\mathbf u}_\varepsilon\), \(\vartheta_\varepsilon\) and \(Y_\varepsilon\) represent the density, the fluid velocity vector, the absolute temperature and the reactant mass fraction, respectively, \({\mathbf j}=(0,0,1)\), the specific entropy \(s=4a\vartheta^4/(3\rho)+S(\rho/\vartheta^{5/2})\), with \(S\) correlated to \(P\), and \(a\) and \(h\) denote some positive constants. If the transport coefficients \(\mu\), \(\kappa\) and \(d\) obey \( 0<c_1(1+\vartheta)\leq \mu(\vartheta),\) \(|\mu'(\vartheta)|\leq c_2;\) \(0<c_1(1+\vartheta^3)\leq \kappa(\vartheta)\leq c_2(1+\vartheta^3);\) \(0<c_1(1+\vartheta)\leq d(\vartheta),\) \( |d'(\vartheta)|\leq c_2,\) for all \(\vartheta>0, \) and the initial conditions are such that \(\rho_\varepsilon(0)=\bar\rho+\varepsilon \rho_{\varepsilon,0}^{(1)}\), \({\mathbf u}_\varepsilon(0)={\mathbf u}_{\varepsilon,0}\), \(\vartheta_\varepsilon(0)=\bar\vartheta+\varepsilon \vartheta_{\varepsilon,0}^{(1)}\), \(Y_\varepsilon(0)=\varepsilon Y_{\varepsilon,0}^{(1)}\) satisfy some additional convergences, the authors study how an asymptotic limit \(({\mathbf u},\Theta,Z)\) can represent a variational solution to a reduced reactive Boussinesq system \[ \bar\rho( \partial_t{\mathbf u}+{\mathbf u}\cdot\nabla{\mathbf u})- \mu(\bar\vartheta)\Delta{\mathbf u} =-\nabla p-\bar\rho\bar\alpha\Theta{\mathbf j};\qquad\nabla\cdot {\mathbf u}=0; \] \[ \bar\rho\bar c_p( \partial_t\Theta+{\mathbf u}\cdot\nabla\Theta) =\kappa({\bar\theta})\Delta\Theta; \qquad\bar\rho(\partial_t Z+{\mathbf u}\cdot\nabla Z)=d(\bar\vartheta)\Delta Z, \] in a bounded tube-like domain \(\Sigma \times]0,1[\subset\Omega\subset\Sigma\times\mathbb R\), of class \(C^3\), with \(\Sigma\) denoting a bounded domain of \(\mathbb R^2\), for \(p\in \mathcal{D}'((0,T)\times\Omega)\) and some constants \(\bar\alpha\) and \(\bar c_p\). The proof of convergence is based on the finding of adequate estimates.