×

Stability of coexisting phases for compressible van der Waals fluids. (English) Zbl 0787.35065

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 \((v-\overline v,u- \overline u,e-\overline e)\).

MSC:

35Q30 Navier-Stokes equations
76T99 Multiphase and multicomponent flows
35D05 Existence of generalized solutions of PDE (MSC2000)
35B35 Stability in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI