Well-posedness of mild solutions to the drift-diffusion and the vorticity equations in amalgam spaces (Q2109019)

The author studies parabolic equations with nonlocal quadratic nonlinearities such as the vorticity equation (related to the Navier-Stokes system) and equations for interacting particles (arising in chemotaxis, astrophysics and electrochemistry). In order to take into account possible solutions with a slow decay at infinity, the functional framework of the amalgam spaces is considered. In those spaces the behavior of the \(L^p\) Lebesgue norms is controlled over the whole in \(\mathbb{R}^n\) in the \(L^\nu\) sense. In particular, usual \(L^p\) and uniform local \(L^p\) spaces can be considered. The main results include local-in-time existence of solutions with the initial data in suitable amalgam spaces, global-in-time existence for small initial data, and some unconditional uniqueness results.