Hadamard well-posedness for a class of nonlinear shallow shell problems

John Cagnol (1) , Irena Lasiecka (2) , Catherine Lebiedzik (3) , Richard Marchand (4)

1 ESILV - École Supérieure d'Ingénierie Léonard de Vinci
2 University of Virginia
3 Wayne State University [Detroit]
4 Department of Mathematics

This paper is concerned with the nonlinear shallow shell model introduced in 1966 by W.T. Koiter in [On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B 69 (1966) 33–54, Section 11] and later studied in [M. Bernadou, J.T. Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60(3) (1981) 285–308]. We consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zolésio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard well-posedness of weak solutions, as shown in this paper, is a new result which solves an old open problem in the field.

Mots clés: Intrinsic geometric shell modeling; Hadamard well-posedness; Hyperbolic partial differential equation; Koiter nonlinear shell model

Domaines: Mathématiques [math]

John Cagnol, Irena Lasiecka, Catherine Lebiedzik, Richard Marchand. Hadamard well-posedness for a class of nonlinear shallow shell problems. Nonlinear Analysis: Theory, Methods and Applications, 2007, 67 (8), pp.2452-2484. ⟨10.1016/j.na.2006.09.004⟩. ⟨hal-01570319⟩