## 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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### References:

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