KdV on an incoming tide

Given smooth step-like initial data V(0, x) on the real line, we show that the Korteweg–de Vries equation is globally well-posed for initial data $u(0,x)\in V(0,x)+{H}^{-1}(\mathbb{R})$. The proof uses our general well-posedness result (2021 arXiv:2104.11346). As a prerequisite, we show that KdV is globally well-posed for ${H}^{3}(\mathbb{R})$ perturbations of step-like initial data. In the case V ≡ 0, we obtain a new proof of the Bona–Smith theorem (Bona and Smith 1975 Trans. R. Soc. A 278 555–601) using the low-regularity methods that established the sharp well-posedness of KdV in H⁻¹ (Killip and Vişan 2019 Ann. Math. 190 249–305).