Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities

• Published in: Calculus of variations and partial differential equations 2016-08, Vol.55 (4), p.1-39, Article 107
• Main Authors: Cao, Xinru, Lankeit, Johannes
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Fluid dynamics; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Smooth boundaries; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-016-1027-2

The coupled chemotaxis fluid system n t = Δ n - ∇ · ( n S ( x , n , c ) · ∇ c ) - u · ∇ n , ( x , t ) ∈ Ω × ( 0 , T ) , c t = Δ c - n c - u · ∇ c , ( x , t ) ∈ Ω × ( 0 , T ) , u t = Δ u - ( u · ∇ ) u + ∇ P + n ∇ Φ , ∇ · u = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , ∇ c · ν = ( ∇ n - n S ( x , n , c ) · ∇ c ) · ν = 0 , u = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , n ( x , 0 ) = n 0 ( x ) , c ( x , 0 ) = c 0 ( x ) , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , where S ∈ ( C 2 ( Ω ¯ × [ 0 , ∞ ) 2 ) ) N × N , is considered in a bounded domain Ω ⊂ R N , N ∈ { 2 , 3 } , with smooth boundary. We show that it has global classical solutions if the initial data satisfy certain smallness conditions and give decay properties of these solutions.