Exponential Growth and Properties of Solutions for a Forced System of Incompressible Navier–Stokes Equations in Sobolev–Gevrey Spaces

One problem of interest in the analysis of Navier–Stokes equations is concerned with the behavior of solutions for certain conditions in the forcing term or external force. In this work, we consider an external force of a maximum exponential growth, and we investigate the local existence and uniqueness of solutions to the incompressible Navier–Stokes equations within the Sobolev–Gevrey space H a , σ 1 ( R 3 ) . Sobolev–Gevrey spaces are well suited for our purposes, as they provide high regularity and control over derivative growth, and this is particularly relevant for us, given the maximum exponential growth in the forcing term. Additionally, the structured bounds in Gevrey spaces help monitor potential solution blow-up by maintaining regularity, though they do not fully prevent or resolve global blow-up scenarios. Utilizing the Banach fixed-point theorem, we demonstrate that the nonlinear operator associated with the Navier–Stokes equations is locally Lipschitz continuous in H a , σ 1 ( R 3 ) . Through energy estimates and the application of Grönwall’s inequality, we establish that solutions exist, are unique, and also exhibit exponential growth in their Sobolev–Gevrey norms over time under certain assumptions in the forcing term. This analysis in intended to contribute in the understanding of the behavior of fluid flows with forcing terms in high-regularity function spaces.