Nucleophilic substitution in solvolysis reaction of tert-butyl bromide goes through [tex]S_{N}1[/tex] mechanism.
Because Br is a good leaving group due to it's large size and high polarizability and methanol is an weak nucleophile.
In the first step, tert-butyl cation is produced. In the second step, methanol attacks the carbocation and produce methyl tert-butyl ether after deprotonation.
Reaction and structure of product has been shown below.
