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Global wellposedness of KdV in \(H^{-1}(\mathbb T,\mathbb R)\). (English) Zbl 1106.35081

In the two following theorems the global in time well-posedness in Sobolev space \(H^{\beta}(\mathbb{T},\mathbb{R}),\;\; \beta \geq -1\) (\(\mathbb{T}\) is the circle \(\mathbb{R}/\mathbb{Z}\) of the length \(1\)) of the Cauchy problem for the KdV equation \[ \partial_tv(x,t)=-\partial_x^3v(x,t)+6v(x,t)\partial_x v(x,t), v(0,x)=q \in H^{\beta}(\mathbb{T}) \] is proved.
(1) KdV is globally \(C^0\)-wellposed in \(H^{\beta}(\mathbb{T})\) for any \(\beta\geq -1.\) In particular, the solution map \(\mathcal{J}:H^{\beta}(\mathbb{T})\to C(\mathbb{R},H^{\beta}(\mathbb{T}))\) has the group property \(\mathcal{J}(t+s,q)=\mathcal{J}(t,\mathcal{J}(s,q)),\;\forall t,s \in \mathbb{R}\) and for any \(t\in \mathbb{R}\;\; \mathcal{J}_t:H^{\beta}(\mathbb{T})\to H^{\beta}(\mathbb{T}),\; q\longmapsto \mathcal{J}(t,q)\) is a homomorphism.
(2) For any \(q\in H^{\beta}(\mathbb{T})\) with \(\beta\geq -1\) the solution \(t\longmapsto \mathcal{J}(t,q)\) has the following properties:
(i) \(\{\mathcal{J}(t,q)|t\in \mathbb{R}\}\)is relatively compact in \(H^{\beta}(\mathbb{T})\);
(ii) the solution \(t\longmapsto \mathcal{J}(t,q)\) is almost periodic;
(iii) \(\mathcal{J}(t,q)\in I_{s0}(L_q)=\{p\in H^{-1}(\mathbb{T})|\text{spec}(L_p)=\text{spec}(L_q)\},\) with \(L_q=-\frac{d^2}{dx^2}+q\) being the Hill operator and spec\((L_q)\) denoting the periodic spectrum of \(L_q\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
35G25 Initial value problems for nonlinear higher-order PDEs
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