Bridging the difference between Fourier's law and Navier-Stokes equations

Thermal transport often deviates from Fourier's law and in hydrodynamic conditions, as for example in layered and two-dimensional materials, heat flux more closely resembles the flow of liquids rather than the diffusive flow predicted by Fourier's law. Here, we start from the linearized phonon Boltzmann transport equation to discuss how hydrodynamic heat transport in crystals arises from the propagation of both energy and crystal momentum fluxes. Microscopically, the two fluxes are distinguished by the parity of relaxons (eigenvectors of the scattering matrix), with odd relaxons contributing to energy transport, and even relaxons contributing to momentum transport. Macroscopically, the parity symmetry gives rise to two coupled equations, which we term viscous heat equations, describing thermal transport in terms of fields of temperature and phonon drift velocity. The energy and momentum diffusion is controlled by the coefficients of thermal conductivity and viscosity respectively. These viscous heat equations, which can be parametrized with ab-initio simulations, extend Fourier's law to the hydrodynamic regime and represent the thermal counterpart of Navier-Stokes equations in the linear and laminar regime, extending the reach of mesoscopic models of heat conduction.