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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.

MSC:

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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