Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

Abstract

We continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the case of positive energy. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining two different frequency regimes. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. We combine the linear estimates with the Fourier decomposition method and X s,b spaces to obtain local well-posedness of NV at positive energy in H s , s > 1 2 . Our result implies, in particular, that at least for s > 1 2 , NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili equations.As a supplement, we provide some new explicit solutions of NV at zero energy which exhibit an interesting behaviour at large times.

Dispersive Estimates for Full Dispersion KP Equations

“…This is not a simple matter since the symbol p and p defined in (1.3) and (1.4) are non-homogeneous and also non-polynomial. Similar difficulties occur for other non-standard dispersive equation such as the Novikov-Veselov equation [10] or a higher dimensional version of the Benjamin-Ono equation [8].…”

Low regularity local well-posedness for the zero energy Novikov-Veselov equation

“…For KdV the reduction of the general to the mean zero case [8, p. 219] is trivial in the sense that it leaves the L 4estimate and the resonance function unchanged. For (NV) this reduction produces the additional linear term With E = 3φ 0 this is precisely the situation of the "nonzero energy" (NV) analyzed in [19,20] in the nonperiodic case. The resonance function is then disturbed by the additional term and the exact cancellation of the Fourier multiplier is destroyed.…”

“…To treat the endpoint case he used the U p -and V p -spaces introduced by Koch and Tataru [17,25,26]. In [19,20] Kazeykina and Munoz generalized the s > 1 2 result mentioned above to the more general "nonzero energy NV equation", which is much harder to analyze. All these LWP results rely exclusively on a global smoothing effect of solutions to the linear part of the equation, expressed in terms of (eventually bilinear) Strichartz-type estimates with derivative gain.…”

Numerical study of blow-up and stability of line solitons for the Novikov–Veselov equation

“…Though the local existence theory for NV equation has been developed, there is little hope to obtain global results via the use of the conservation laws. For E = 0 an explicit construction confirms the fact that the L 2 norm cannot be bounded by conservation laws: for NV at E = 0 in [KM2] the authors construct examples of solutions which blow up in L 2 norm in infinite time. Define…”