The heat equation under conditions on the moments in higher dimensions - Université Polytechnique des Hauts-de-France Accéder directement au contenu
Article Dans Une Revue Mathematical News / Mathematische Nachrichten Année : 2015

The heat equation under conditions on the moments in higher dimensions

Résumé

We consider the heat equation on the N-dimensional cube (0, 1)N and impose different classes of integral conditions, instead of usual boundary ones.Well-posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of N = 1, for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general N the heat equation with such integral conditions is still well-posed, upon suitably relaxing the notion of solution. Existence of solutions with general initial data in a suitable space of distributions over (0, 1)N are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained turns out to solve the heat equation only in a certain distributional sense. However, one of these realizations is tightly related to a well-known object of operator theory, the Krein–von Neumann extension of the Laplacian. This connection also establishes well-posedness in a classical sense, as long as the initial data are L2-functions.
Fichier principal
Vignette du fichier
1302.5529.pdf (209.45 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03163722 , version 1 (05-07-2022)

Identifiants

Citer

Delio Mugnolo, Serge Nicaise. The heat equation under conditions on the moments in higher dimensions. Mathematical News / Mathematische Nachrichten, 2015, 288 (2-3), pp.295-308. ⟨10.1002/mana.201300298⟩. ⟨hal-03163722⟩
11 Consultations
13 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More