Local well-posedness for the modified KdV equation in almost critical ^H^r_s-spaces

Axel Gruenrock; Luis Vega

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Local well-posedness for the modified KdV equation in almost critical ^H^r_s-spaces

Axel Gruenrock·Luis Vega

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Abstract

We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<ξ>^s u^_0||_{L^r'_ξ}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, so the case (s,r)=(0,1), which is critical in view of scaling considerations is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation.

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Axel Gruenrock, Luis Vega. 2007-03-01. Local well-posedness for the modified KdV equation in almost critical ^H^r_s-spaces. https://arxiv.org/abs/math/0703037

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