Nonuniqueness and Uniqueness in the Initial-Value Problem for Burgers’ Equation

Source abstract

A sharp local existence and uniqueness theory for the initial-value problem for Burgers’ equation is given in the Sobolev spaces $H^s$, ${{ - 1} / 2} < s \leq 0$. It is proved that these results cannot be extended to any $s < {{ - 1} / 2}$ because uniqueness fails. A particular nontrivial solution is found which converges to 0 in the $H^s$-norm as $t \to 0^ +$.