-\mu_0\partial_t\mathbf{H},
\end{array}
\right.$$ where $\mathbf{H}$ and ${\mathbf{E}}$ are respectively the magnetic and electric fields, $ \varepsilon_0$ is the permittivity constant, $\mu_0$ is the permeability constant, $\varepsilon_{\text{loc}}=\varepsilon_{\infty}+\varepsilon_{\text{inter}}$ is introduced to account for the local response, and ${\mathbf{J}}$ is the nonlocal hydrodynamic polarization current density which is due to the nonlocal material on the plasmonic scatterers [@HiremathEtal2012]. In this paper, we will for simplicity set $\varepsilon_{\text{inter}}=0$ and $\varepsilon_{\infty}=1$, thereby focusing solely on the free-electron response to light. Equations need to be completed to solve electromagnetic fields ${\mathbf{E}}$ and $\mathbf{H}$ because of the unknown polarization current density ${\mathbf{J}}$. The models that we will consider in this paper differ only in the assumed dynamics of the polarization current density, which we will now discuss in more detail.

The polarization current density ${\mathbf{J}}$ due to the motion of the free-electron gas can be written as $$\label{eq:jHD}
{\mathbf{J}}= -en{\mathbf{v}},$$ where $e$ is the charge of the electron, $n$ is the density of the electron gas (a scalar field), and ${\mathbf{v}}$ is its hydrodynamic velocity (a vector field). Within the hydrodynamic model, the dynamics of the velocity field is given by [@Schmittetal2016; @RazaEtal2015] $$\label{eq:hd}
m_e(\partial_t+{\mathbf{v}}\cdot\nabla){\mathbf{v}}=-e({\mathbf{E}}+{\mathbf{v}}\times\mathbf{B}) - m_e\gamma{\mathbf{v}}-\nabla\left(\frac{\delta g[n]}{\delta n}\right),$$ where $m_e$ is the mass of an electron, 