What does Islamic law obligate me to do regarding my mother who goes out in immodest clothing and does not accept advice, and is it permissible for me to abandon her in this situation?

The son must continue to gently advise his mother who disobeys Allah's law, because gentleness adorns advice and increases the likelihood of a positive response, especially with a mother whose dutifulness is affirmed.

If the advice is not effective and only increases her stubbornness, then it must be abandoned, based on Allah's saying: "So remind, if the reminder benefits."

It is recommended to have some virtuous individuals advise her.

Do not forget to frequently supplicate for her guidance.

It is not permissible to abandon one's parents due to their disobedience, except within very narrow limits, because it usually does not benefit and may even increase their stubbornness.