  We study the time evolution of a one-dimensional interacting fermion system
described by the Luttinger model starting from a nonequilibrium state defined
by a smooth temperature profile $T(x)$. As a specific example we consider the
case when $T(x)$ is equal to $T_L$ ($T_R$) far to the left (right). Using a
series expansion in $\epsilon = 2(T_{R} - T_{L})/(T_{L}+T_{R})$, we compute the
energy density, the heat current density, and the fermion two-point correlation
function for all times $t \geq 0$. For local (delta-function) interactions, the
first two are computed to all orders, giving simple exact expressions involving
the Schwarzian derivative of the integral of $T(x)$. For nonlocal interactions,
breaking scale invariance, we compute the nonequilibrium steady state (NESS) to
all orders and the evolution to first order in $\epsilon$. The heat current in
the NESS is universal even when conformal invariance is broken by the
interactions, and its dependence on $T_{L,R}$ agrees with numerical results for
the $XXZ$ spin chain. Moreover, our analytical formulas predict peaks at short
times in the transition region between different temperatures and show
dispersion effects that, even if nonuniversal, are qualitatively similar to
ones observed in numerical simulations for related models, such as spin chains
and interacting lattice fermions.
