On two coupled degenerate parabolic equations motivated by thermodynamics. (English) Zbl 1512.35393
Summary: We discuss a system of two coupled parabolic equations that have degenerate diffusion constants depending on the energy-like variable. The dissipation of the velocity-like variable is fed as a source term into the energy equation leading to conservation of the total energy. The motivation of studying this system comes from Prandtl’s and Kolmogorov’s one- and two-equation models for turbulence, where the energy-like variable is the mean turbulent kinetic energy. Because of the degeneracies, there are solutions with time-dependent support like in the porous medium equation, which is contained in our system as a special case. The motion of the free boundary may be driven by either self-diffusion of the energy-like variable or by dissipation of the velocity-like variable. The crossover of these two phenomena is exemplified for the associated planar traveling fronts. We provide existence of suitably defined weak and very weak solutions. After providing a thermodynamically motivated gradient structure, we also establish convergence into steady state for bounded domains and provide a conjecture on the asymptotically self-similar behavior of the solutions in \(\mathbb{R}^d\) for large times.
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 49S05 | Variational principles of physics |
| 80M30 | Variational methods applied to problems in thermodynamics and heat transfer |
Keywords:
degenerate parabolic system; free-boundary problem; convergence into thermal equilibrium; gradient structure; mass and momentum conservation; energy-dissipation estimates; solutions with time-dependent supportReferences:
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