Non-unique solutions to the periodic gKdV equation

Abstract

In this paper we utilize a convex integration scheme to construct non-trivial weak solutions to the $k$-generalized KdV equation which lie in $$ \bigcap_{ε> 0} C_t^0 L_x^{k-ε}([0,1] \times \mathbb{T}) $$ and, when $k \ge 3$, it may also be chosen in \[ \bigcap_{ε>0} C_t^0 H_x^{\frac{1}{2} - \frac{1}{k} - ε}([0,1] \times \mathbb{T}) \] attaining identically $0$ initial data. Since our solutions do not lie in $C_t^0 L_x^k$, this requires introducing a new notion of weak solution, which is in fact stronger than the classical notion of a weak solution when the nonlinearity is integrable. This result shows that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C_t^0L^1_x$. In the case of KdV this is in fact also sufficient.