"Argue" is a common synonym for "art one's case", meaning to present one's side of an argument or to persuade someone to agree with your viewpoint. Some antonyms for this phrase could include "yield", "concede", or "surrender". These words imply that instead of presenting a strong argument or fighting for one's beliefs, one is willingly giving up or accepting defeat. Other possible antonyms could be "ignore" or "disregard", indicating a lack of interest or attention on the topic at hand. Ultimately, the choice of antonym depends on the context in which the phrase is being used and the intended meaning of the speaker.