  The existence of Ulrich modules for local domains has been a difficult and
elusive open question. For over thirty years, it was unknown whether local
domains always have Ulrich modules. In this paper, we answer the question of
existence for both Ulrich modules and weakly lim Ulrich sequences -- a weaker
notion recently introduced by Ma -- in the negative. We construct many local
domains in all dimensions $d \geq 2$ that do not have any Ulrich modules.
Moreover, we show that when $d = 2$, these local domains do not have weakly lim
Ulrich sequences. A key insight in our proofs is the classification of MCM
$R$-modules via the $S_2$-ification of $R$. For local domains of dimension $2$,
we show that the existence of weakly lim Ulrich sequences implies the existence
of lim Ulrich sequences.
