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On the Cauchy problem for the Benjamin-Ono equation. (English) Zbl 0608.35030

We analyse the Cauchy problem for the Benjamin-Ono (BO) equation namely \[ \partial_ tu=-\partial_ x(u^ 2+2\sigma \partial_ xu),\quad u(x,0)=\phi (x), \] in various function spaces. Here \(u=u(x,t)\), \(x\in {\mathbb{R}}\), \(t>0\) while \(\sigma\) denotes the Hilbert transform. Using parabolic regularization (i.e. adding \(\mu \partial^ 2_ xu\) to the R.H.S. of BO, solving with \(\mu >0\) and taking limits as \(\mu\downarrow 0)\) we establish existence (local in time) and uniqueness in the Sobolev spaces \(H^ s({\mathbb{R}})\), \(s>3/2\). Global existence is proved for \(s\geq 2\) using Case’s conserved quantities and norm compression (a method originally due to T. Kato). Global existence is also obtained in \({\mathcal F}_ 2=L^ 2_ 2({\mathbb{R}})\cap H^ 2({\mathbb{R}})\) where \({\mathcal F}_ r=L^ 2_ r({\mathbb{R}})\cap H^ 2({\mathbb{R}})\), and \(L^ 2_ r({\mathbb{R}})\) denotes the usual weighted \(L^ 2\) spaces. We show that there are non- trivial solutions in \({\mathcal F}_ 3\) iff \({\hat \phi}\)(0)\(=0\), in which case they are globally defined, while if u(t), \(t\in [0,T]\), \(T>0\) is a solution in \({\mathcal F}_ 4\), it must be zero. This result is a combination of the nonlinearity with the non-smoothness of the Hilbert transform. After the publication of the paper in question we have established global existence, uniqueness and global continuous dependence on the initial condition in the spaces \({\mathcal F}_{2\gamma}=L^ 2_{2\gamma}({\mathbb{R}})\cap H^ 2({\mathbb{R}})\), \(\gamma\in [0,1]\). The main motivation for this result is the fact that the solitons associated with BO live in \({\mathcal F}_ r\), \(r<3/2\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35A35 Theoretical approximation in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35K15 Initial value problems for second-order parabolic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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