Blow-Up of Modified Navier--Stokes Equations in Nonhomogeneous Fourier Spaces

Abstract

We study the three-dimensional incompressible Navier--Stokes equations with a componentwise nonlinear damping term of the form $\alpha\sum_{k=1}^3 u_k^{2m+1}e_k$ in Fourier and Fourier-- Gevrey spaces based on $\mathcal{X}^0(\mathbb{R}^3)$. We first establish local-in-time existence and uniqueness of solutions and obtain the corresponding unique maximal solutions. We then derive blow-up criteria in the Fourier--Gevrey framework, including an integral blow-up criterion in a weaker exponential weight. Moreover, we establish a quantitative lower bound for the solution near a possible finite maximal existence time. By iterating the loss of exponential weight, we finally obtain a corresponding lower bound in the unweighted Fourier space $\mathcal{X}^0(\mathbb{R}^3)$.