APPLICATIONS OF MATHEMATICS, Vol. 50, No. 6, pp. 527-541, 2005

# A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum

## Jens Frehse, Sonja Goj, Josef Malek

J. Frehse, S. Goj, University of Bonn, Institute for Applied Mathematics, Beringstr. 6, 53115 Bonn, Germany, e-mails: erdbeere@iam.uni-bonn.de, goj@mailcip.iam.uni-bonn.de; J. Malek, Mathematical Institute, Charles University, Sokolovska 83, 186 75 Praha 8, Czech Republic, e-mail: malek@karlin.mff.cuni.cz

Abstract: We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities $\rho_i$ of the fluids and their velocity fields $u^{(i)}$ are prescribed at infinity: $\rho_i|_{\infty} = \rho_{i \infty} > 0$, $u^{(i)}|_{\infty} = 0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely $\rho_i \equiv\rho_{i \infty}$, $u^{(i)} \equiv0$, $i=1,2$.

Keywords: miscible mixture, compressible fluid, uniqueness, zero force

Classification (MSC 2000): 35Q30, 76N10

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