## 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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### References:

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