  The properties of a phase at finite interactions can be significantly
influenced by the underlying dispersion of the non-interacting Hamiltonian. We
demonstrate this by studying the repulsive Hubbard model on the $2$D Lieb
lattice, which has a flat band for vanishing interaction $U$. We perform
real-space dynamical mean-field theory calculations at different temperatures
and dopings using a continuous time quantum Monte Carlo impurity solver.
Studying the frequency dependence of the self-energy, we show that a finite
temperature non-magnetic non-Fermi liquid behavior is a concomitant of the flat
band singularity. At half-filling we also find a magnetically ordered region,
where the order parameter varies linearly with the interaction strength, and a
strongly correlated Mott insulating phase. The double occupancy decreases
sharply for small $U$, highlighting the flat band contribution. Away from
half-filling, we observe the stripe order, i.e. an inhomogeneous spin and
charge density wave of finite wavelength which turns into a sub-lattice
ordering at higher temperatures.
