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Rigidity aspects of singular patches in stratified flows

Taoufik Hmidi, Haroune Houamed and Mohamed Zerguine

Vol. 4 (2022), No. 3, 465–557

We explore the local well-posedness theory for the 2D inviscid Boussinesq system when the vorticity is given by a singular patch. We give a significant improvement on the result of Hassainia and Hmidi (J. Math. Anal. Appl. 430:2 (2015), 777–809) by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We establish a uniform LWP theory with respect to the vanishing conductivity. This issue is much more delicate than the inviscid case and one should carefully deal with various difficulties related to the diffusion effects which tend to alter some local structures. The weak a priori estimates are not trivial and refined analysis on transport-diffusion equation subject to a logarithmic singular potential is required. Another difficulty stems from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing effects of transport-diffusion equation advected by a velocity field which scales slightly below the Lipschitz class.

well-posedness, singular patches, smoothing effects
Mathematical Subject Classification
Primary: 35Q35, 76B03, 76B70
Received: 21 November 2021
Accepted: 10 April 2022
Published: 9 November 2022
Taoufik Hmidi
Université de Rennes 1
Haroune Houamed
Department of Mathematics
New York University in Abu Dhabi
Abu Dhabi
United Arab Emirates
Mohamed Zerguine
Faculté des Mathématiques et d’Informatique
Département de Mathématiques
Université de Batna