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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Global existence for $ 1$D, compressible, isentropic Navier-Stokes equations with large initial data


Author: David Hoff
Journal: Trans. Amer. Math. Soc. 303 (1987), 169-181
MSC: Primary 35Q10; Secondary 76D05
MathSciNet review: 896014
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Abstract: We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in $ {L^2}$ and $ {L^2} \cap BV$ respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds $ 3/2$.


References [Enhancements On Off] (What's this?)

  • [1] David Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 301-315. MR 866843 (88b:35156)
  • [2] David Hoff and Tai-Ping Liu, (to appear).
  • [3] David Hoff and Joel Smoller, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. Henri Poincaré, Analyse Non linéaire 2 (1985), 213-235. MR 797271 (87b:35078)
  • [4] A. Kazhikov and V. Shelukhin, Unique global solutions in time of initial boundary value problems for one dimensional equations of a viscous gas, Prikl. Mat. Meh. 41 (1977), 282-291= J. Appl. Math. Mech. 41 (1977), 273-283. MR 0468593 (57:8425)
  • [6] Jong Uhn Kim, Global existence of solutions of the equations of one dimensional thermoviscoelasticity with initial data in $ BV$ and $ {L^1}$, Ann. Scuola Norm. Sup. Pisa 10 (1983), 357-427. MR 739917 (85k:35210)
  • [7] Denis Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, preprint. MR 867555 (88a:35193)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0896014-6
PII: S 0002-9947(1987)0896014-6
Article copyright: © Copyright 1987 American Mathematical Society