Is there a cure for pneumothorax?

Almost always, yes: Depending on the cause, it will either resolve on its own, or it can be treated by evacuating the air in the chest (to expand the lung) or with a chest tube to drain the air while the defect in the lung heals (after which the lung expands to fill the pleural space).

Yes: It may resolve on its own. It may require a catheter or tube to evacuate the air to relieve any shortness of breath. It may need surgery if it recurs to remove blebs (air cysts) at the top of the lung.