One-dimensional compressible heat-conducting gas with temperature-dependent viscosity. (English) Zbl 1353.35241
Summary: We consider the one-dimensional compressible Navier-Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity \(\mu\) and the heat conductivity \(\kappa\) depend on the specific volume \(v\) and the temperature \(\theta\) and are both proportional to \(h(v)\theta^{\alpha}\) for certain non-degenerate smooth function \(h\). We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter \(\alpha\) and initial data, which imply that the initial data can be large if \(|\alpha|\) is sufficiently small. Such a result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
Keywords:
compressible Navier-Stokes system; temperature-dependent viscosity; ideal polytropic gas; global solutions; large initial data; general adiabatic exponentReferences:
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