From Longman Dictionary of Contemporary Englishcurl up phrasal verb1 BENDto move so that you are lying or sitting with your arms and legs bent close to your body I just wanted to curl up and go to sleep. Sarah was curled up on the sofa.2 BENDif something flat curls up, its edges start to become curved and point upwards The letter was now yellow and beginning to curl up. → curl→ See Verb table
Examples from the Corpus
curl up• I was so tired all I wanted to do was curl up and watch TV.• She curled up in her bed that night, thinking of Michel.• The two cats curled up together in the armchair. 