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On the local well-posedness of the Benjamin-Ono equation in \(H^S(\mathbb{R})\). (English) Zbl 1039.35106

The authors prove the following theorem for the Benjamin-Ono equation \[ u_{t}+\mathbf{H}u_{xx}+uu_{x}=0,\;\;u(0,x)=u_{0}(x),\tag{8} \] where \(\mathbf{H}\) denotes the Hilbert transform.
Fix \(s>\frac{5}{4}\). Then for every \(u_{0}\in{\mathbf{H}^{s}(\mathbb{R})}\), there exist \(T\geq{| | u_{0}| | _{\mathbf{H}^{s}}^{-4}}\) and a unique solution of (8) on the time interval \([0,T]\) satisfying \[ u\in{C([0,T],L^{2}(\mathbb{R}))},\;\;u_{x}\in{L^{1}}{([0,T],L^{\infty}(\mathbb{R}))}. \] Moreover, for any \(R>0,\) there exists \(T\geq{R^{-4}}\) such that the nonlinear map \(u_{0}\to{u}\) is continuous from the ball of radius \(R\) of \(\mathbf{H}^{s}(\mathbb{R})\) to \(C([0,T],\mathbf{H}^{s}(\mathbb{R}))\).
Conditions for an improvement of the theorem are given.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
42B25 Maximal functions, Littlewood-Paley theory
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