Convection of Discontinuities in Solutions of the Navier-Stokes Equations for Compressible Flow

• David Hoff
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 29)

Abstract

We report here on results concerning the existence, uniqueness, and continuous dependence on initial data of discontinuous solutions of the Navier-Stokes equations for one-dimensional compressible fluid flow:
$$\left\{ \begin{gathered} {\upsilon_t} - {u_x} = 0 \hfill \\ {u_t} + p{(\upsilon, e)_x} = {\left( {\frac{{ \in {u_x}}}{\upsilon }} \right)_x} \hfill \\ {\left( {\frac{{{u^2}}}{2} + e} \right)_t} + {(up(\upsilon, e))_x} = {\left( {\frac{{ \in u{u_x} + \lambda T{{(\upsilon, e)}_x}}}{\upsilon }} \right)_x} \hfill \\ \end{gathered} \right.$$
(1)
with Cauchy data
$$\left[ \begin{gathered} \upsilon \hfill \\ u \hfill \\ e \hfill \\ \end{gathered} \right](x,0) = \left[ \begin{gathered} {\upsilon_0} \hfill \\ {u_0} \hfill \\ {e_0} \hfill \\ \end{gathered} \right](x)$$
(2)
Here v, u, e, p, and T represent respectively the specific volume, velocity, specific internal energy, pressure, and temperature in a fluid; x is the Lagrangian coordinate, so that x = constant corresponds to a particle path; and and λ are fixed positive viscosity parameters.

Keywords

Entropy Convection

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References

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David Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible now, (to appear in Arch. Rational Mech. Ana).Google Scholar
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David Hoff, Global existence and stability of viscous, nonisentropic flows, (to appear).Google Scholar
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David Hoff and Tai-Ping Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, (to appear in Indiana Univ. Math. J).Google Scholar
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David Hoff and Joel Smoller, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. Henri Poincaré, Analyse Non linéare 2 (1985), 213–235.
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Roger Zarnowski and David Hoff, A finite difference scheme for the Navier-Stokes equations of one-dimensional, isentropic, compressible flow, (to appear).Google Scholar