Keller–Segel–Stokes interaction involving signal‐dependent motilities

Tian, Yu; Winkler, Michael

doi:10.1002/mma.9419

Cited by 3 publications

References 42 publications

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Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing

Fuest,

Winkler

2024

J. Math. Fluid Mech.

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The chemotaxis-Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{rcl} n_t+u\cdot \nabla n & =& \Delta \big (n c^{-\alpha } \big ), \\ c_t+ u\cdot \nabla c & =& \Delta c -nc,\\ u_t + (u\cdot \nabla ) u & =& \Delta u+\nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \end{aligned}$$ n t + u · ∇ n = Δ ( n c - α ) , c t + u · ∇ c = Δ c - n c , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain $$\Omega \subset \mathbb R^2$$ Ω ⊂ R 2 . For all $$\alpha > 0$$ α > 0 and all sufficiently regular $$\Phi $$ Φ , we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for u at a point in the proof when knowledge about n is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time $$L^p$$ L p estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time $$L^2$$ L 2 norm of the force term raised to an arbitrary small power.

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