Fundamental solutions for parabolic equations and systems: universal existence, uniqueness, representation - Laboratoire de Mathématiques d'Orsay
Pré-Publication, Document De Travail Année : 2024

Fundamental solutions for parabolic equations and systems: universal existence, uniqueness, representation

Pascal Auscher
Khalid Baadi
  • Fonction : Auteur
  • PersonId : 1482362

Résumé

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational definition. Our classes of weak solutions are taken with minimal assumptions. We prove the existence and uniqueness of a fundamental solution which seems new in this generality: it is shown to always coincide with the associated evolution family for the initial value problem with zero source and it yields representation of all weak solutions. Our strategy is a variational approach avoiding density arguments, a priori regularity of weak solutions or regularization by smooth operators. One of our main tools are embedding results which yield time continuity of our weak solutions going beyond the celebrated Lions regularity theorem and that is addressing a variety of source terms. We illustrate our results with three concrete applications : second order uniformly elliptic part with Dirichlet boundary condition on domains, integro-differential elliptic part, and second order degenerate elliptic part.
Fichier principal
Vignette du fichier
Fundamental solutions for parabolic equations and systems (Hal).pdf (559.95 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
licence

Dates et versions

hal-04857120 , version 1 (27-12-2024)

Licence

Identifiants

  • HAL Id : hal-04857120 , version 1

Citer

Pascal Auscher, Khalid Baadi. Fundamental solutions for parabolic equations and systems: universal existence, uniqueness, representation. 2024. ⟨hal-04857120⟩
3 Consultations
0 Téléchargements

Partager

More