In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation ∂tv − bH∂ xv + a∂ xv = cv∂xv − d∂x(vH∂xv + H(v∂xv)), where x, t ∈ R, v is a real-valued function, H is the Hilbert transform, a ∈ R, b, c and d are positive constants, is locally well-posed for initial data v(0) = v0 ∈ H(R), s ≥ 2 or v0 ∈ H(R) ∩ L(R; xdx), k ∈ Z+, k ≥ 2.

Janice L. Wong, Robert F. Higgins, Indrani Bhowmick, David Xi Cao, Géza Szigethy, Joseph W. Ziller, Matthew P. Shores* and Alan F. Heyduk* A new bimetallic platform comprising a six-coordinate Fe(ONO)2 unit bound to an (ONO)M (M1⁄4 Fe, Zn) has been discovered ((ONO)H3 1⁄4 bis(3,5-di-tert-butyl-2-phenol)amine). Reaction of Fe(ONO)2 with either (ONO)Fe(py)3 or with (ONO )FeCl2 under reducing cond...

Considering the Cauchy problem for the modified finite-depthfluid equation ∂tu− Gδ(∂ 2 xu)∓ u 2ux = 0, u(0) = u0, where Gδf = −iF [coth(2πδξ)− 1 2πδξ ]Ff , δ&1, and u is a real-valued function, we show that it is uniformly globally well-posed if u0 ∈ Hs (s ≥ 1/2) with ‖u0‖L2 sufficiently small for all δ&1. Our result is sharp in the sense that the solution map fails to be C in Hs(s < 1/2). More...

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation ∂tu+ |∂x| ∂xu+ uux = 0, u(x, 0) = u0(x), is locally well-posed in the Sobolev spaces H for s > 1 − α if 0 ≤ α ≤ 1. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru [13] to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and T...

We consider the Benjamin–Ono equation on torus with an additional damping term smallest Fourier modes ( $$\cos $$ and $$\sin ). first prove global well-posedness of this in $$L^2_{r,0}(\mathbb {T})$$ . Then, we describe weak limit points trajectories when time goes to infinity, show that these are strong points. Finally, boundedness higher-order Sobolev norms for equation. Our key tool is Birkh...