Uniqueness of Solutions for the Generalized Korteweg–de Vries Equation

Source abstract

The uniqueness of $L^2$ and $H^1$ solutions of the Cauchy problem for the generalized Korteweg–de Vries (KdV) equation $$\partial _t u + D^3 u + a(u)Du = 0$$ with $a \in \mathcal{C}(\mathbb{R},\mathbb{R})$ and with initial data $u_0$ in weighted $L^2$ or $H^1$ spaces according to $$\left( {1 + x_ + } \right)^{{\beta / 2}} u_0 \in L^2 ,\qquad \left( {1 + x_ + } \right)^{{\gamma / 2}} Du_0 \in L^2$$ is studied for some $\beta ,\gamma \geqq 0$. Several uniqueness classes are exhibited, and the a priori estimates are derived for the corresponding norms of smooth solutions in terms of the initial data required to implement compactness existence proofs of solutions in those classes. For (weighted) $L^2$ solutions, the results given here cover the case where $| {a(\rho )} | \leqq C| \rho |^p$ with $0 < p < \frac{7}{2}$ , $\beta = {1 / p} - \frac{1}{4}$ if $p \leqq 2$ and $\beta = \frac{1}{4}$ if $p \geqq 2$. For the ordinary KdV equation with $p = 1$, the result $\beta = \frac{3}{4}$ improves over previously known results by a factor of 2. For $H^1$ solutions, uniqueness and a priori estimates with initial data $u_0 \in H^1$ (namely, $\beta = \gamma = 0$) are proved provided $p > \frac{3}{2}$ . For the ordinary KdV equation with $p = 1$, the results given here yield uniqueness and a priori estimates for$\beta + \gamma \geqq \frac{1}{2}$ (for instance, $\beta = \frac{1}{2}$, $\gamma = 0$, or $\beta ({3 / {16}})$, $\gamma = ({5 / {16}})$.