Global strong solution with vacuum to the two dimensional density-dependent Navier-Stokes system (Q2878980)

The authors study the Navier-Stokes system for a two dimensional incompressible fluid. They assume that the viscosity changes depending on the density, and they also allow the existence of the vacuum (i.e., the density may be equal to 0). Denoting the viscosity \(\mu\) as a function \(\mu (\rho )\) of the density \(\rho\), they consider the Navier-Stokes equation \((\rho u)_t +\mathrm{div}\!\;(\rho u\otimes u) -\mathrm{div}\!\;(2\mu (\rho )d)+\nabla P=0\). Here \(u,\;d\), and \(P\) denote the velocity, the deformation tensor, and the pressure, respectively. Y.~Cho and H.~Kim considered the corresponding initial value problem with initial values \(\rho_0 \in W^{1,q}\) and \(u_0 \in H^1_{0,\sigma }\cap H^2\) with \(q>2\) [\textit{Y. Cho} and \textit{H. Kim}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 59, No. 4, 465--489 (2004; Zbl 1066.35070)], and proved that if the solution has a finite time span \(T^*\), then either \(||\nabla \rho (t)||_{L^q}\) or \(||\nabla u(t)||_{L^2}\) must blow-up at \(t=T^*\). However, the present article proves that in case of a finite time span, \(||\nabla \rho (t)||_{L^p}\) must blow-up at \(t=T^*\) for \(2<p\leq q\). In particular, if \(\mu\) is a constant, then the solution always exists globally. A global existence theorem for the strong solution is already known if the density is strictly positive, and the authors extend this result allowing the presence of vacuum.