| Abstract: |
| The mathematical modeling of the flow of chemically reactive mixtures gas been gaining increasing interest the recent years. We investigate the model described in the monograph of V. Giovangigli. The model is composed of compressible Navier-Stokes equations coupled with a system of reaction-diffusion equations desrcibing the evolution of fractional masses.
The main difficulty lies in the lack of symmetry of the cross-diffusion matrix. For this reason, the existence results for the system require usually certain simplifying assumptions such as diagonality of the cross-diffusion matrix (Fick Law) or assumption of equal molar masses.
More recently, the global existence of weak solutions admitting general diffusion matrix and different molar masses has been obtained by Mucha, Pokorny and Zatorska. However, for strong solutions the only global existence results were available in the whole space, assuming that solutions are close to equilibrium.
In my talk I would like to present our recent result with Y. Shibata and E. Zatorska on the existence of strong solutions in a bounded domain for a two component flow under constant temperature, however for a non-diagonal cross-diffusion matrix and different molar masses.
The proof consist in rewriting the problem in new unknowns which lead to a symmetric system. Then we apply the $L_p$-$L_q$ maximal regularity theory and estimates for the nonlinearities in order to obtain local in time solutions. Finally, using appropriate exponential decay estimate we are able to prolong this solutions for arbitrary times under additional smallness assumptions. The result can be considered a first step in the proof of analogous results for a multispecies model with variable temperature. |
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