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Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. (English) Zbl 1231.35180

The authors consider the Cauchy problem for the 2D Boussinesq system

t v +v ·v -·ν(θ)v +p=θe , t θ+v ·θ-·κ(θ)θ=0,·v =0,v ,θ t=0 =v 0 ,θ 0 ,(1)

where e =(0,1), v =(v 1 ,v 2 ) is the velocity, p is the pressure, the kinematic viscosity ν is a positive function satisfying

C 0 -1 ν(θ)C 0 ,

and the diffusivity coefficient κ is is a positive function satisfying

C 0 -1 κ(θ)C 0 ·

They prove that, provided that θ 0 H s and v 0 is a divergence-free (H s ) 2 -vector-field with s>2, system (1) admits a unique global solution such that

v C( + ;(H s ) 2 )L 2 ( + ;(H s+1 ) 2 ),

and

θC( + ;H s )L 2 ( + ;H s+1 ),

for any T>0.

MSC:
35Q35PDEs in connection with fluid mechanics
76D03Existence, uniqueness, and regularity theory
35B30Dependence of solutions of PDE on initial and boundary data, parameters
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