Uniform Lp Estimates for Solutions to the Inhomogeneous 2D Navier-Stokes Equations and Application to a Chemotaxis-Fluid System with Local Sensing

The chemotaxis-Navier-Stokes system { n(t)+u & sdot;del n=Delta(nc(-alpha)), c(t)+u & sdot;del c=Delta c-nc, u(t)+(u & sdot;del)u=Delta u+del P+n del Phi, del & sdot;u=0, modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain Omega subset of R-2. For all alpha>0 and all sufficiently regular Phi, 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 estimates for solutions to the inhomogeneous Navier-Stokes equations merely depending on the space-time L-2 norm of the force term raised to an arbitrary small power. We're sorry, something doesn't seem to be working properly. Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.