Contractivity and Asymptotics in Wasserstein Metrics for Viscous Nonlinear Scalar Conservation Laws

Carrillo, José A. ; Di Francesco, Marco ; Lattanzio, Corrado

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 2, pp. 277-292.

Abstract (VO)

In this work, recent results concerning the long time asymptotics of one- dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-de- pendent asymptotic profiles for a large class of convection-diffusion problems with fully general nonlinearities and with degenerate diffusion.

Riassunto

In questo articolo sono riportati alcuni risultati recenti riguardo il comportamento asintotico nel tempo di leggi di conservazione scalari in una dimensione spaziale e con densità di probabilità come dati iniziali. Tali risultati sono quindi applicati al caso di leggi di conservazione viscose con diffusioni nonlineari degeneri. Le proprietà di contrazione nella distanza di trasporto massimale e di uniforme espansione delle soluzioni forniscono l'esistenza di profili asintotici dipendenti dal tempo per un'ampia classe di equazioni di convenzione-diffusione con nonlinearità arbitrarie e diffusione degenere.

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