Time periodic problem for the compressible Navier-Stokes equation on \(\mathbb{R}^2\) with antisymmetry (Q1743702)

The author deals with the following system of equations \[ \begin{cases} \partial_t\rho + \operatorname{div} (\rho v)=0, \\ \rho(\partial_tv+(v\cdot \nabla)v)-\mu \Delta v -(\mu +\mu^{\prime})\nabla \operatorname{div} v + \nabla p(\rho)=\rho g, \end{cases} \] where \(\rho = \rho (x,t)\) denotes the density, \(v=(v_1(x,t),v_2(x,t))\) represents the velocity field, \(t\geq 0\), \(x\in \mathbb{R}^2\); \(p=p(\rho)\) is the pressure; \(\mu\) and \(\mu^{\prime}\) are the viscosity coefficients, \(\mu \geq 0\), \(\mu + \mu^{\prime}\geq 0\); and \(g=g(x,t)\) represents an external force, \[ g(x,t+T)=g(x,t), \;\;x\in \mathbb{R}^2, \;t\in \mathbb{R}. \] The existence of a \(T\)-periodic solution is proven for sufficiently small \(g\) with antisymmetry condition. The proof is based on using the \(T\)-map associated with the linearized problem around the stationary solution with constant density.