On Convolution in Variable Lebesgue Spaces and Applications to Fractional Navier_Stokes Equations

Abstract

In this paper, we introduce a new class of convolution-type inequalities in variable exponent Lebesgue spaces and derive several related estimates, including the \(L^{r(\cdot)}\)--\(L^{p(\cdot)}\) smoothing estimate for the fractional heat kernel. We demonstrate the usefulness of these inequalities by establishing the local well-posedness results for mild solutions to the fractional Navier--Stokes equations, and we further extend these results to global-in-time well-posedness for sufficiently small initial data. Our analysis is carried out in a wide range of mixed-norm variable exponent Lebesgue spaces, including the fully variable setting $L^{p(\cdot)}_t L^{q(\cdot)}_x$, highlighting the robustness of the proposed framework under non-constant integrability. Moreover, the proposed framework is expected to serve as a key tool for similar applications in other related variable exponent function spaces.