On well-posedness for some Korteweg–de Vries type equations with variable coefficients

Abstract

In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from zero and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution u such that hu belongs to a classical Sobolev space, where h is a function related to this ratio. The LWP in $H^s({\mathbb {R}})$, $s>1/2$, in the classical (Hadamard) sense is also proven under this time an assumption of boundedness of the above primitive function. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s({\mathbb {R}})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.

Appendix

1.1 Proof of Lemma 2.5

Let $N>0$. We follow [10]. By Plancherel and the mean-value theorem,

$$\begin{aligned} \Bigl | ( [P_N, P_{\ll N}f] g)(x)\Bigr |&=\Bigl |( [P_N, P_{\ll N}f] \widetilde{P}_N g)(x)\Bigr | \\&=\Bigl | \int _{{\mathbb {R}}} {{\mathcal {F}}}^{-1}_x(\varphi _N)(x-y) P_{\ll N}f(y) \widetilde{P}_N g(y) \, dy \\&\hspace{1cm} - \int _{{\mathbb {R}}} P_{\ll N}f(x) {{\mathcal {F}}}^{-1}_x(\varphi _N)(x-y) \widetilde{P}_N g(y) \, dy\Bigr | \\&= \Bigl | \int _{{\mathbb {R}}}(P_{\ll N}f (y) -P_{\ll N}f(x)) N {{\mathcal {F}}}^{-1}_x(\varphi )(N(x-y)) \widetilde{P}_N g(y) \, dy\Bigr | \\&\le \Vert P_{\ll N}f_x\Vert _{L^\infty _x}\int _{{\mathbb {R}}} N |x-y| |{{\mathcal {F}}}^{-1}_x(\varphi )(N(x-y))| |\widetilde{P}_N g(y)| \, dy \end{aligned}$$

Therefore, since $N |\cdot | |{{\mathcal {F}}}^{-1}_x(\varphi )(N \cdot )|=|{{\mathcal {F}}}^{-1}_x(\varphi ')(N \cdot ) |$ we deduce from Young’s convolution inequalities that

$$\begin{aligned} \Bigl \Vert [P_N, P_{\ll N}f] g)\Vert _{L^2}&\lesssim N^{-1} \Vert P_{\ll N}f_x\Vert _{L^\infty _x} \Vert \widetilde{P}_N g\Vert _{L^2} \; , \end{aligned}$$

that is (2.13).