Data di Pubblicazione:
2017
Citazione:
Generalized solutions in PDEs and the Burgers' equation / V. Benci, L. Luperi Baglini. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 263:10(2017), pp. 6916-6952. [10.1016/j.jde.2017.07.034]
Abstract:
In many situations, the notion of function is not sufficient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDEs, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDEs, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
PDEs; Generalized solutions; Burgers' equation; Nonstandard analysis
Elenco autori:
V. Benci, L. Luperi Baglini
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