Well-posedness and small data scattering for the generalized Ostrovsky equation

Abstract

We consider the generalized Ostrovsky equation u tx = u + (u p ) xx . We show that the equation is locally well posed in H s , s > 3/2 for all integer values of p 2. For p 4, we show that the equation is globally well posed for small data in H 5 ∩ W 3,1 and moreover, it scatters small data. The latter results are corroborated by numerical computations which confirm the heuristically expected decay of u L r ∼ t −(r−2)/(2r) .

Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation

“…Local well-posedness of the Cauchy problem for the reduced Ostrovsky equations (1.1) and (1.2) can be shown inḢ s per with s > 3 2 [16,21]. For sufficiently large initial data, the local solutions break in finite time, similar to the inviscid Burgers equation [5,10,16].…”

Large time asymptotics of solutions to the short-pulse equation

“…The crucial point is that the decay of the remainder part is faster than t − 1 2 , which is the decay rate of the free solution. In fact, Stefanov et al proved the dispersive estimate [17]). Setting p = ∞ formally, we expect the decay rate of the L ∞ norm of the free solution to be t − 1 2 .…”

“…Pelinovsky and Sakovich [15] showed global wellposedness in the energy space for p = 3 and small initial data. Stefanov et al [17] showed local existence of a unique solution to (1.2) with u 0 ∈ H s (R) when s > 3 2 . They also proved global existence and scattering for p ≥ 4 and small initial data u 0 ∈ H 5 (R) ∩ W 3,1 (R).…”

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

“…In the early 90's, Vakhnenko, [24] proposed an alternative derivation, while Hunter, [11] proposed some numerical simulations. The well-posedness questions were investigated by Boyd, [1]; Schaefer and Wayne, [21]; Stefanov-Shen-Kevrekidis, [22]. Liu, Pelinovsky and Sakovich, [16,17] have studied wave breaking, which was later supplemented by the global regularity results for small, in appropriate sense data, of GrimshawPelinovsky, [7].…”