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**Continuous dependence in \(L^ 2\) for discontinuous solutions of the viscous \(p\)-system.**
*(English)*
Zbl 0836.35157

Summary: We prove that discontinuous solutions of the Navier-Stokes equations for isentropic or isothermal flow depend continuously on their initial data in \(L^2\). This improves earlier results in which continuous dependence was known only in a much stronger norm, a norm inappropriately strong for the physical model. We also apply our continuous dependence theory to obtain improved rates of convergence for certain finite difference approximations.

### MSC:

35R05 | PDEs with low regular coefficients and/or low regular data |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

### Keywords:

discontinuous solutions of the Navier-Stokes equations; continuous dependence; rates of convergence; finite difference approximations
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\textit{D. Hoff} and \textit{R. Zarnowski}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, No. 2, 159--187 (1994; Zbl 0836.35157)

### References:

[1] | Donoghue, W. F., Distributions and Fourier transforms, (1969), Academic Press · Zbl 0188.18102 |

[2] | Hoff, D., Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Royal Soc. Edinburgh, Sect., A103, 301-315, (1986) · Zbl 0635.35074 |

[3] | Hoff, D., Global existence for ID, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., Vol. 303, No. 11, 169-181, (1987) · Zbl 0656.76064 |

[4] | Hoff, D., Discontinuous solutions of the Navier-Stokes equations for compressible flow, Archive Rational. Mech. Ana., Vol. 114, 15-46, (1991) · Zbl 0732.35071 |

[5] | Hoff, D., Global well-posedness of the Cauchy problem for nonisentropic gas dynamics with discontinuous initial data, J. Differential Equations, Vol. 95, No. 1, 33-73, (1992) · Zbl 0762.35085 |

[6] | Hoff, D.; Smoller, J., Error bounds for glimm difference approximations for scalar conservation laws, Trans. Amer. Math. Soc., Vol. 289, 611-642, (1985) · Zbl 0579.65096 |

[7] | Kuznetsov, N. N., Accuracy of some approximate methods for computing the weak solutions of a first-order quasilinear equation, Zh. Vychisl. Math. Fiz., Vol. 16, 1489-1502, (1976) · Zbl 0354.35021 |

[8] | Zarnowski, R.; Hoff, D., A finite difference scheme for the Navier-Stokes equations of compressible, isentropic flow, SI AM J. on Numer. Anal., Vol. 28, 78-112, (1991) · Zbl 0727.76094 |

[9] | Zarnowski, R., Existence, uniqueness, and computation of solutions for mixed problems in compressible flow, J. Math. Anal, and Appl., Vol. 169, No. 2, 515-545, (1992) · Zbl 0777.35063 |

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