Sharp ill-posedness results for the KdV and mKdV equations on the torus (Q436130)
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scientific article; zbMATH DE number 6060956
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Sharp ill-posedness results for the KdV and mKdV equations on the torus |
scientific article; zbMATH DE number 6060956 |
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Sharp ill-posedness results for the KdV and mKdV equations on the torus (English)
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30 July 2012
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KdV equation
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mkdv equation
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ill-posedness
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weak solution
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0.89919376
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0.8936075
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0.8898517
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0.88782674
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0.88780904
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0.8837405
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0.87608445
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In this paper the author considers the Cauchy problem for the Korteweg-de Vries (KdV) NEWLINE\[NEWLINE w_t + w_{xxx} - 6w w_x = 0 NEWLINE\]NEWLINE and the modified KdV (mKdV) equation NEWLINE\[NEWLINE v_t + v_{xxx} \mp 6v^2 v_x = 0 NEWLINE\]NEWLINE on the flat torus \(\mathbb T = \mathbb R\setminus 2\pi\mathbb Z\). Two main results are proved:NEWLINENEWLINE1. The discontinuity of the solution-map associated with the KdV and the mKdV equations in \(H^s(\mathbb T)\) for \(s < -1\) and \(s < 0\) respectively.NEWLINENEWLINE2. Existence of weak \(L^2\)-solutions of the mKdV equation.NEWLINENEWLINEIt also constructs global weak solutions of the renormalized mKdV equation.
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