WIAS Preprint No. 2937, (2022)

On two coupled degenerate parabolic equations motivated by thermodynamics


  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2020 Mathematics Subject Classification

  • 35K65 35K40 80M30 49S05


  • Free boundary, growing support, gradient system, energy conservation, momentum conservation, porous medium equation, energy-dissipation estimates, entropy estimates




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 cross-over 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 Rd for large times.

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