Well-posedness for KdV-type equations with quadratic nonlinearity
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Journal of Evolution Equations
Well-posedness for KdV-type equations with quadratic nonlinearity Hiroyuki Hirayama, Shinya Kinoshita and Mamoru Okamoto
Abstract. We consider the Cauchy problem of the KdV-type equation ∂t u +
1 3 ∂ u = c1 u∂x2 u + c2 (∂x u)2 , u(0) = u 0 . 3 x
Pilod (J Differ Equ 245(8):2055–2077, 2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space H s (R) for any s ∈ R if c1 = 0. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in H 2 (R) with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in H 1 (R) with bounded primitives.
1. Introduction We consider the Cauchy problem for the Korteweg–de Vries (KdV)-type equation 1 ∂t u + ∂x3 u = c1 u∂x2 u + c2 (∂x u)2 , 3
(1)
where u is a real valued function and c1 and c2 are real constants. If c1 = 0, because ∂x u satisfies the KdV equation, the results by Kenig et al. [13] and Kishimoto [8] imply that (1) is well-posed in the Sobolev space H s (R) for s ≥ 41 . On the other hand, Tarama [24] proved that even a linear equation requires a Mizohata-type condition for the well-posedness in L 2 (R) (see also [18]). Indeed, the linear equation (∂x + ∂x3 + a(x)∂x2 )u = 0 where a is smooth with bounded derivatives is well-posed in L 2 (R) if and only if x2 sup a(x)dx < ∞ x1 ≤x2
x1
holds. Hence, at least, well-posedness in H s (R) for (1) requires some additional conditions. In fact, Pilod [21] showed that the flow map of this Cauchy problem fails to be twice differentiable in H s (R) for any s ∈ R if c1 = 0. Mathematics Subject Classification: 35Q53, 35A01 Keywords: KdV-type equation, Well-posedness, Gauge transformation.
J. Evol. Equ.
H. Hirayama et al.
Local well-posedness was established using the weighted Sobolev spaces H s (R) ∩ L 2 (x 2k d x) for sufficiently large s and k by Kenig et al. [12] and Kenig and Staffilani [14]. For the proof, they used a change of dependent variables as in [6,7]. In these works, the change of dependent variable was called a gauge transformation. By replacing weighted spaces with a spatial summability condition, Harrop-Griffiths [4] proved local well-posedness for (1) in a translation invariant space l 1 H s (R) for s > 25 . We note that he also treated more general semi-linear nonlinearity (see also [5]). Here, l 1 H s (R) is defined as follows: We denote the set of nonnegative integers by N0 := N ∪ {0}. For each N ∈ 2N0 , we take a partition Q N of R into intervals of length N and an associated smooth partition of unity χQ , 1= Q∈Q N
where we assume χ Q ∼ 1 on Q and supp χ Q ⊂ {x ∈ R : |x − y| < 21 , y ∈ Q}. Then, we define u l 1 L 2 = χ Q u L 2 . N
Q∈Q N
We define the space l 1 H s (R) with the norm ⎛ u l 1 H s
=⎝
⎞1 2
N
2s
N ∈N0
PN u l21 L 2 ⎠ 2
,
N
where PN is the Littlewood–Paley projection defined in Sect. 1.1. We mention the well-posedness results for the third-order Benjamin–Ono equation ∂t u − bH∂x
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