Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data. (English) Zbl 0762.35085

The author considers the Cauchy problem for the nonisentropic, compressible Navier-Stokes equations in one space dimension, i.e. \[ v_ t-u_ x=0,\quad u_ t+p(v,e)_ x=\left({\varepsilon u_ x\over v}\right)_ x\quad\text{in }\mathbb{R}\times(0,\infty), \]
\[ \left(e+{u^ 2\over 2}\right)_ t+(up(v,e))_ x=\left({\varepsilon uu_ x+\lambda T(v,e)_ x\over v}\right)_ x,\quad (v,u,e)(0)=(v_ 0,u_ 0,e_ 0)\quad\text{in }\mathbb{R}. \] Under suitable conditions on the data of the problem he proves the existence of a global weak solution in a neighborhood of a constant state \((\tilde v,\tilde u,\tilde e)\). Furthermore he closely studies the qualitative behavior of the solution, e.g. the evolution of jump discontinuities is examined and the convergence of the solution against \((\tilde v,\tilde u,\tilde e)\) as \(t\to\infty\) is shown. Under the additional condition that the gas is ideal the author is able to prove that the solution is unique and that it continuously depends on the initial data.


35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35R05 PDEs with low regular coefficients and/or low regular data
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[1] Hoff, D., 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 · Zbl 0635.35074
[2] Hoff, D., Global existence for \(1D\), compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303, 169-181 (1987) · Zbl 0656.76064
[3] Hoff, D., Discontinuous solutions of the Navier-Stokes equations for compressible flow, Arch. Rational Mech. Anal., 114, 15-46 (1991) · Zbl 0732.35071
[4] Hoff, D.; Liu, Tai-Ping, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., 38, No. 4, 861-915 (1989) · Zbl 0674.76047
[5] Kanel, Y., On a model system of equations of one-dimensional gas motion, Differential Equations, 4, 374-380 (1968) · Zbl 0235.35023
[6] Kazhikov, A.; Shelukhin, V., Unique global solutions in time of initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Mech., 41, 273-283 (1977)
[7] Kim, Jong Uhn, 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, No. 3, 357-427 (1983) · Zbl 0534.73011
[8] Serre, D., Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C.R. Acad. Sci. Paris Sér. I, 303, No. 13 (1986) · Zbl 0597.76067
[9] Serre, D., Sur l’équation monodimensionelle d’un fluide visqueux, compressible et conducteur de chaleur, C.R. Acad. Sci. Paris Sér. I, 303, No. 14 (1986)
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