Resonance and generalized Navier-Stokes equations (Q2721938)

The author considers the incompressible Navier-Stokes system NEWLINE\[NEWLINEu_t-\varepsilon \Delta u+(u\cdot \nabla)u -\nabla p=f(t,x,u)+h, \quad \text{ div} u =0NEWLINE\]NEWLINE on the \(n\)-dimensional torus (\(n\geq 2\)). One of the main technical assumptions on the external force \(f\) says that its \(j\)th component depends on the \(j\)th component of the unknown velocity \(u(x,t)\). Moreover, \(u_jf_j(t,x,u_j)\) behaves like \(|u_j|^k\) for some \(1\leq k\leq 2+4/n\). It is proved that for \(1\leq k\leq 2\), the initial value problem for this Navier-Stokes system has a global weak solution. In addition, if \(2<k<2+4/n\), this problem is shown to have a weak solution on some time interval.