On the torus, the fifth-order KdV-BBM model is locally and globally well-posed in H^s for s>=1, and fails to be well-posed with norm inflation for s<1.
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On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain
On the torus, the fifth-order KdV-BBM model is locally and globally well-posed in H^s for s>=1, and fails to be well-posed with norm inflation for s<1.