A bilinear Airy-estimate with application to gKdV-3

The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem ut + uxxx + (u 4)x = 0, u(0) = u0 ∈ H s x(R), s > − 1 6 for the generalized Korteweg-deVries equation of order three, for short gKdV3. For real valued data u0 ∈ L 2 x(R) global wellposedness follows by the conservation of the L-norm. The main new tool is a bilinear estimate for solutions of the Airy-equation. The purpose of this note is to establish local wellposedness of the CauchyProblem ut + uxxx + (u )x = 0, u(0) = u0 ∈ H s x(R), s > − 1 6 for the generalized Korteweg-deVries equation of order three, for short gKdV-3. So far, local wellposedness of this problem is known for data u0 ∈ H s x(R), s ≥ 1 12 . This was shown by Kenig, Ponce and Vega in 1993, see Theorem 2.6 in [KPV93]. Here we extend this result to data u0 ∈ H s x(R), s > − 1 6 . A standard scaling argument suggests that this is optimal (up to the endpoint). For real valued data u0 ∈ L 2 x(R) we obtain global wellposedness by the conservation of the L-norm. By the Fourier restriction norm method introduced in [B93] and further developed in [KPV96] and [GTV97] matters reduce to the proof of the estimate ‖∂x ∏4 i=1 ui‖Xs,b′ ≤ c 4 ∏ i=1 ‖ui‖Xs,b (1) for suitable values of s, b and b. Here the space Xs,b is the completion of the Schwartz class S(R) with respect to the norm ‖u‖Xs,b = ( ∫ dξdτ < τ − ξ >< ξ > |Fu(ξ, τ)| ) 1 2 , where F denotes the Fourier transform in both variables. The main new tool for the proof of (1) is the following bilinear Airy-estimate: 1 Lemma 1 Let I denote the Riesz potential of order −s and let I −(f, g) be defined by its Fourier transform (in the space variable): FxI s −(f, g)(ξ) := ∫ ξ1+ξ2=ξ dξ1|ξ1 − ξ2| Fxf(ξ1)Fxg(ξ2). Then we have ‖I 1 2 I 1 2 −(e −t∂3u1, e 3u2)‖L2xt ≤ c‖u1‖L2x‖u2‖L2x . Proof: We will write for short û instead of Fxu and ∫