## 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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