On Well-posedness and stability for a Keller-Segel-Navier-Stokes system in Lorentz spaces over real hyperbolic spaces

Abstract

We establish the well-posedness and exponential stability of mild solutions for the Patlak-Keller-Segel-Navier-Stokes system on the real hyperbolic space \(\mathbb {H}^d(\mathbb {R})\, (d\geqslant 2)\). We consider the system obtained by coupling a parabolic-elliptic Keller–Segel-type model with a Navier–Stokes-type equation. First, We employ the \(L^p-L^q\) estimates for both scalar and vectorial heat semigroups to derive useful interpolation inequalities, including \(L^{p,r}-L^{q,r}\) estimates and Yamazaki-type estimates on \(\mathbb {H}^d(\mathbb {R})\). Using these interpolation estimates, we establish the global well-posedness of mild solutions for the corresponding linear systems in the framework of Lorentz space \(L^{p,r}(\mathbb {H}^d)\) for \(d<p<2d\) and \(1\leqslant r \leqslant \infty \). Then, we combine the existence results of linear systems with fixed-point arguments to obtain the well-posedness of mild solutions for the semilinear systems. In addition, we prove an exponential stability for mild solutions by using again the interpolation estimates and the Gronwall inequality. Finally, we also consider the well-posedness and stability for the systems in the weak-\(L^p\) spaces, i.e., \(L^{p,\infty }(\mathbb {H}^d(\mathbb {R}))\) for the case \(d\geqslant 3\).