Abstract
Abstract
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
)
∈
V
(
0
,
x
)
+
H
−
1
(
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
(
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).
Funder
National Science Foundation
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Cited by
3 articles.
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