Generalized solutions in PDEs and the Burgers' equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Benci, Vieri^1,2  Luperi Baglini, Lorenzo³ 

[1] Univ Pisa, Dipartimento Matemat, Via F Buonarroti 1-C, I-56127 Pisa, Italy

[2] Ctr Linceo Interdisciplinare Beniamino Segre, Palazzo Corsini Via Lungara 10, I-00165 Rome, Italy

[3] Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

关键词: PDEs;    Generalized solutions;    Burgers' equation;    Nonstandard analysis;   

DOI  :  10.1016/j.jde.2017.07.034

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free