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

Conference paper

## 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:
with Cauchy data
Here

$$ \left\{ \begin{gathered} {\upsilon_t} - {u_x} = 0 \hfill \\ {u_t} + p{(\upsilon, e)_x} = {\left( {\frac{{ \in {u_x}}}{\upsilon }} \right)_x} \hfill \\ {\left( {\frac{{{u^2}}}{2} + e} \right)_t} + {(up(\upsilon, e))_x} = {\left( {\frac{{ \in u{u_x} + \lambda T{{(\upsilon, e)}_x}}}{\upsilon }} \right)_x} \hfill \\ \end{gathered} \right. $$

(1)

$$ \left[ \begin{gathered} \upsilon \hfill \\ u \hfill \\ e \hfill \\ \end{gathered} \right](x,0) = \left[ \begin{gathered} {\upsilon_0} \hfill \\ {u_0} \hfill \\ {e_0} \hfill \\ \end{gathered} \right](x) $$

(2)

*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

Compressible Flow Continuous Dependence Discontinuous Solution Inviscid Flow Cauchy Data
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [1]David Hoff,
*Discontinuous solutions of the Navier-Stokes equations for compressible now*, (to appear in Arch. Rational Mech. Ana).Google Scholar - [2]David Hoff,
*Global existence and stability of viscous, nonisentropic flows*, (to appear).Google Scholar - [3]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 - [4]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.MathSciNetMATHGoogle Scholar - [5]Roger Zarnowski and David Hoff,
*A finite difference scheme for the Navier-Stokes equations of one-dimensional, isentropic, compressible flow*, (to appear).Google Scholar

## Copyright information

© Springer-Verlag New York, Inc. 1991