Must my ex-advisor be a co-author in new research after the defense? When writing papers about new research made from my doctoral research results, after my defense, must my doctoral advisor be included as co-author?
The traditional role of authorship is to clarify who (significantly) contributed to the work being published. Doctoral advisors often contribute actively to the research being performed, and thus are included as co-authors on that particular regard.

However, they should not be included as co-authors automatically. Only if they have actively contributed towards the paper's ideas, implementation and/or writing. The least that can be expected is that they contributed to the paper's ideas (during your supervision) and that they have read and edited the paper being published. 

That said, it is usually expected that you invite contributors to co-write a paper, if they have contributed significant ideas. If your doctoral advisor did that for the research you made during your PhD studies, then you should invite them for those particular papers.