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Hmidi, Taoufik; Houamed, Haroune; Zerguine, Mohamed

Rigidity aspects of singular patches in stratified flows. (English) Zbl 1501.35316

Tunis. J. Math. 4, No. 3, 465-557 (2022).
Summary: 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 Z. Hassainia and T. Hmidi [J. Math. Anal. Appl. 430, No. 2, 777–809 (2015; Zbl 1319.35189)] 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B70 Stratification effects in inviscid fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q31 Euler equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Keywords:

well-posedness; singular patches; smoothing effects

Citations:

Zbl 1319.35189
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\textit{T. Hmidi} et al., Tunis. J. Math. 4, No. 3, 465--557 (2022; Zbl 1501.35316)
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