Does the fast of one who intended to fast every Monday of each month become valid if he drank forgetfully after the dawn prayer (Fajr), and is a new intention required for each day?

Your fast is invalid, as you did not form the intention [of fasting] and you drank after Fajr. You are not required to make up for it unless your intention for that fast was to fulfill a vow to fast on Mondays of every month; in that case, you must fast one day in its place. However, if you intended to fast and then drank forgetfully, your fast is valid and nothing is incumbent upon you.