Is it permissible for the young brother, who lives with his elder brother and is controlled by the latter, to sever ties with his elder brother after he insulted him and expelled him from the house, and to declare his independence?

There is no harm in a brother leaving his sibling's home to live independently, even if there is no stated reason for it. However, boycotting him is impermissible; it is one of the major sins and a cause for curse and God's wrath, as Allah states: "So would you, perhaps, if you turned away, cause corruption in the land and sever your ties of kinship? Those are the ones whom Allah has cursed, so He made them deaf and blinded their vision." You must maintain ties with your brother. It is preferable to advise him and clarify his error with wisdom. There is no harm in seeking mediation from relatives and righteous people.