  In this paper, we discuss the general existence theory of Dirac-harmonic maps
from closed surfaces via the heat flow for $\alpha$-Dirac-harmonic maps and
blow-up analysis. More precisely, given any initial map along which the Dirac
operator has nontrivial minimal kernel, we first prove the short time existence
of the heat flow for $\alpha$-Dirac-harmonic maps. The obstacle to the global
existence is the singular time when the kernel of the Dirac operator no longer
stays minimal along the flow. In this case, the kernel may not be continuous
even if the map is smooth with respect to time. To overcome this issue, we use
the analyticity of the target manifold to obtain the density of the maps along
which the Dirac operator has minimal kernel in the homotopy class of the given
initial map. Then, when we arrive at the singular time, this density allows us
to pick another map which has lower energy to restart the flow. Thus, we get a
flow which may not be continuous at a set of isolated points. Furthermore, with
the help of small energy regularity and blow-up analysis, we finally get the
existence of nontrivial $\alpha$-Dirac-harmonic maps ($\alpha\geq1$) from
closed surfaces. Moreover, if the target manifold does not admit any nontrivial
harmonic sphere, then the map part stays in the same homotopy class as the
given initial map.
