Abstract
In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct \({\mathcal{N}_0}\) of infinite length in \({\mathbb{R}^2}\) with width W 0 and consider two uniform subsonic flow \({{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}\) with different horizontal velocity in \({\mathcal{N}_0}\) divided by a flat contact discontinuity \({\Gamma_{cd}}\). And, we slightly perturb the boundary of \({\mathcal{N}_0}\) so that the width of the perturbed duct converges to \({W_0+\omega}\) for \({|\omega| < \delta}\) at \({x=\infty}\) for some \({\delta >0 }\). Then, we prove that if the asymptotic state at left far field is given by \({{U_l}^{\pm}}\), and if the perturbation of boundary of \({\mathcal{N}_0}\) and \({\delta}\) is sufficiently small, then there exists unique asymptotic state \({{U_r}^{\pm}}\) with a flat contact discontinuity \({\Gamma_{cd}^*}\) at right far field(\({x=\infty}\)) and unique weak solution \({U}\) of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to \({{U_l}^{\pm}}\) and \({{U_r}^{\pm}}\) at \({x=-\infty}\) and \({x=\infty}\) respectively. For that purpose, we establish piecewise C 1 estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of \({\partial\mathcal{N}_0}\) and \({\delta}\).
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Bae, M. Stability of contact discontinuity for steady Euler system in infinite duct. Z. Angew. Math. Phys. 64, 917–936 (2013). https://doi.org/10.1007/s00033-012-0271-3
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DOI: https://doi.org/10.1007/s00033-012-0271-3