Approximation of entropy solutions to degenerate nonlinear parabolic equations

作者：Abreu Eduardo^*; Colombeau Mathilde; Panov Evgeny Yu

来源：Zeitschrift fur Angewandte Mathematik und Physik, 2017, 68(6): 133.

DOI：10.1007/s00033-017-0877-6

• Partial differential equations
• Degenerate parabolic equations
• Entropy solutions
• Approximate solutions
• Stability

We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space , whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and -stability. We prove that the sequence of approximate solutions is strongly -precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.