Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T22:02:33.067Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data

Published online by Cambridge University Press:  14 November 2011

David Hoff
David Hoff
Affiliation:
Department of Mathematics, Indiana University, Swain Hall East, Bloomington, Indiana 47405, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 3-4 , 1986 , pp. 301 - 315
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Hoff, David and Smoller, Joel. Solutions in the large for certain nonlinear parabolic systems. Analyse Nonlinéaire Ann. Inst. H. Poincaré 2 (1985), 213–235.CrossRefGoogle Scholar
2Kanel, Ya.. On a model system of equations of one-dimensional gas motion. Differential Equations 4 (1968), 374–380.Google Scholar
3Kazhikov, A. and Shelukhin, V.. Unique global solutions in time of initial boundary value problems for one dimensional equations of a viscous gas. Appl. Math. Mech. 41 (1977), 273–283.CrossRefGoogle Scholar
4Kim, Jong Uhn. Global existence of solutions of the equations of one dimensional thermoviscoelasticity with initial data in BV and L1. Ann. Scuola Norm. Sup. Pisa X 3 (1983), 357–427.Google Scholar
5Nash, J.. Le problem de Cauchy pour des équations difféientielles d'un fluide générale. Bull. Soc. Math. France1 90 (1962), 487–497.CrossRefGoogle Scholar
6Smoller, Joel. Shock waves and reaction-diffusion equations (New York: Springer, 1983).CrossRefGoogle Scholar