Geodesic
On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation
Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 676-684

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An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coefficient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a finite number of variables. A similar result is true for Stefan's problem. In the limit, when the number of phases tends to infinity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.
Keywords: degenerate nonlinear parabolic equation, Riemann problem, Stefan problem, weak solution, self-similar solution.
Mots-clés : phase transition 
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     author = {E. Yu. Panov},
     title = {On the structure of weak solutions of the {Riemann} problem for a degenerate nonlinear diffusion equation},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {676--684},
     publisher = {mathdoc},
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     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2023_69_4_a7/}
}
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E. Yu. Panov. On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation. Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 676-684. http://geodesic.mathdoc.fr/item/CMFD_2023_69_4_a7/