  We present a nearly-linear time algorithm for finding a minimum-cost flow in
planar graphs with polynomially bounded integer costs and capacities. The
previous fastest algorithm for this problem is based on interior point methods
(IPMs) and works for general sparse graphs in $O(n^{1.5}\text{poly}(\log n))$
time [Daitch-Spielman, STOC'08]. Intuitively, $\Omega(n^{1.5})$ is a natural
runtime barrier for IPM-based methods, since they require $\sqrt{n}$
iterations, each routing a possibly-dense electrical flow.
  To break this barrier, we develop a new implicit representation for flows
based on generalized nested-dissection [Lipton-Rose-Tarjan, JSTOR'79] and
approximate Schur complements [Kyng-Sachdeva, FOCS'16]. This implicit
representation permits us to design a data structure to route an electrical
flow with sparse demands in roughly $\sqrt{n}$ update time, resulting in a
total running time of $O(n\cdot\text{poly}(\log n))$.
  Our results immediately extend to all families of separable graphs.
