Geodesic
Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics
Thomas Alazard1 orcid ; Didier Bresch2 orcid
1Université Paris-Saclay, Gif-sur-Yvette, France
2Université Savoie Mont-Blanc, Le Bourget du Lac, France
Interfaces and free boundaries, Tome 26 (2024) no. 1, pp. 1-30

Voir la notice de l'article provenant de la source EMS Press

DOI

This paper is motivated by the study of Lyapunov functionals for the Hele-Shaw and Mullins-Sekerka equations describing free surface flows in fluid dynamics. We prove that the L2-norm of the free surface elevation and the area of the free surface are Lyapunov functionals. The proofs combine exact identities for the dissipation rates with functional inequalities. We introduce a functional which controls the L2-norm of three-half spatial derivative. Under a mild smallness assumption on the initial data, we show that the latter quantity is also a Lyapunov functional for the Hele-Shaw equation, implying that the area functional is a strong Lyapunov functional. Precise lower bounds for the dissipation rates are established, showing that these Lyapunov functionals are in fact entropies.
DOI : 10.4171/ifb/504
Classification : 35-XX, 76-XX
Mots-clés : free surface, Hele-Shaw, Mullins-Sekerka, Lyapunov, entropy

Thomas Alazard  1   ; Didier Bresch  2

1 Université Paris-Saclay, Gif-sur-Yvette, France
2 Université Savoie Mont-Blanc, Le Bourget du Lac, France 
Thomas Alazard; Didier Bresch. Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics. Interfaces and free boundaries, Tome 26 (2024) no. 1, pp. 1-30. doi: 10.4171/ifb/504
@article{10_4171_ifb_504,
     author = {Thomas Alazard and Didier Bresch},
     title = {Functional inequalities and strong {Lyapunov} functionals for free surface flows in fluid dynamics},
     journal = {Interfaces and free boundaries},
     pages = {1--30},
     year = {2024},
     volume = {26},
     number = {1},
     doi = {10.4171/ifb/504},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/504/}
}
TY  - JOUR
AU  - Thomas Alazard
AU  - Didier Bresch
TI  - Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics
JO  - Interfaces and free boundaries
PY  - 2024
SP  - 1
EP  - 30
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4171/ifb/504/
DO  - 10.4171/ifb/504
ID  - 10_4171_ifb_504
ER  - 
%0 Journal Article
%A Thomas Alazard
%A Didier Bresch
%T Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics
%J Interfaces and free boundaries
%D 2024
%P 1-30
%V 26
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/504/
%R 10.4171/ifb/504
%F 10_4171_ifb_504

Cité par Sources :