Enhanced Existence Time of Solutions to the Fractional Korteweg--de Vries Equation

Stability of traveling waves for the Burgers-Hilbert equation

“…Thanks to the effect of the Hilbert transform and using the normal form method, Hunter, Ifrim, Tataru and Wong (see [12] and [13]) were able to improve this time of existence. More precisely, if is the size of the initial data, they prove a lifespan T ∼ 1 2 for small enough (see also [9] for a similar approach with a modified version of the (BH) equation). The proofs are based in the normal form method and in the modified energy method.…”

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

“…It was proven in [2] that the lifespan of solutions to (5.1) with −1 < α < 1, α = 0 and with initial data of size O(ǫ) in H N (R), N ≥ 3 is O(1/ǫ 2 ). This result is specially striking when −1 < α < 0 since for α > 0 one expects global existence of small solutions.…”

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

“…Some previous results on long time behavior or finite time blow-up for solutions to fractional KdV type equations concerned equations with quadratic nonlinearities, namely the fractional KdV equation (fKdV)∂ t u − |D| α ∂ x u = −u∂ x u, −1 < α < 0,(85)or the Whitham equation ∂ t u + L∂ x u = −u∂ x u. (86) It was proven in[5] that the lifespan of solutions to (85) with −1 < α < 1, α = 0 and with initial data of size O( ) inH N (R), N ≥ 3 is O(1/ 2 ).This result is specially striking when −1 < α < 0 since for α > 0 one expects global existence of small solutions. Note that for the inviscid Burgers equation this lifespan is O(1/ ).Proving a global existence result of small solutions of the fKdV equation in the range −1 < α < 0 or for the Whitham equation (and for the modified Whitham equation) is a challenging open question.…”