"maintain correspondence" is correct and usable in written English.
You can use it when you want to express that you are conducting ongoing written communication with someone. For example, "I maintain correspondence with my grandmother every week by sending her a letter."
He was available enough to maintain correspondence, yet invisible enough to allow for some invention.
A well-known challenge in computable general equilibrium (CGE) models is to maintain correspondence between the forecasted economic and physical quantities over time.
Many alumni classes elected secretaries to maintain correspondence with classmates.
I maintain correspondence with quite a number of people inside the country -- some journalists, some who work in ministries, some students.
The attempt to maintain correspondence between model and physiology has resulted into a large number of parameters.
The Provost also maintained correspondence and subject files for many individual faculty members.
Although forbidden by the pope to continue the controversy with the religious orders, William maintained correspondence with his colleagues at Paris, who subsequently revived the polemic.
After the atrocities committed under Stalin, Schapiro became disillusioned with politics, yet he continued his admiration for the political and maintained correspondence with political figures such as Leon Trotsky.
–49% have maintained correspondence with someone living abroad.
While H.C. "Cliff" Westermann spent much of his mature period in rural Connecticut, he maintained correspondence with artists in California and influenced the careers of a younger generation of artists known as the Chicago Imagists.
He also maintained correspondence with the Jacobites.
