"Live down" is a phrase that means to overcome or outlive a mistake or embarrassment. Some potential synonyms for "live down" could include "overcome," "outlast," "survive," "endure," "weather," "tolerate," "absorb," "withstand," "outshine," or "triumph over." Each of these words captures the sense of moving past a challenging situation or mistake, whether by enduring it until it passes or by actively working to overcome it. Depending on the context and tone of the writing, any of these words could be an effective substitute for "live down" to add variation and nuance to the phrasing.