Hoff, David; Khodja, M. Stability of coexisting phases for compressible van der Waals fluids. (English) Zbl 0787.35065 SIAM J. Appl. Math. 53, No. 1, 1-14 (1993). The authors consider the Navier-Stokes equations for compressible van der Waals fluids in one space dimension, i.e. \[ \begin{aligned} v_ t-u_ x & = 0, \\ u_ t+p(v,e)_ x & =\left( {\varepsilon u_ x \over v} \right)_ x, \\ \left( {1\over 2} u^ 2+e \right)_ t+(up(v,e))_ x & =\left( {\varepsilon uu_ x \over v} +{\lambda T(v,e)_ x \over v} \right)_ x. \end{aligned} \tag{*} \] They examine the nonlinear stability of steady state solutions \((\overline v,\overline u,\overline e)\) consisting of two constant states separated by a convecting phase boundary. It is proved that \((*)\) has a global weak solution \((v,u,e)\) which converges uniformly to \((\overline v,\overline u,\overline e)\) as \(t \to \infty\) provided its initial values are sufficiently close to \((\overline v,\overline u,\overline e)\). The main part of the proof consists in the derivation of time independent upper bounds for various norms of \((v-\overline v,u- \overline u,e-\overline e)\). Reviewer: K.Deckelnick (Freiburg i.Br.) Cited in 6 Documents MSC: 35Q30 Navier-Stokes equations 76T99 Multiphase and multicomponent flows 35D05 Existence of generalized solutions of PDE (MSC2000) 35B35 Stability in context of PDEs Keywords:nonisentropic flow; upper bounds for norms; Navier-Stokes equations; compressible van der Waals fluids; one space dimension; nonlinear stability; steady state solutions; global weak solution PDFBibTeX XMLCite \textit{D. Hoff} and \textit{M. Khodja}, SIAM J. Appl. Math. 53, No. 1, 1--14 (1993; Zbl 0787.35065) Full Text: DOI