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.
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.
Reviewer: Thomas Ernst (Uppsala)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 42B25 | Maximal functions, Littlewood-Paley theory |
