The rod equation is ill-posed in H^s(R) for 1 < s < 3/2 by norm inflation from explicit smooth initial data that is arbitrarily small but produces arbitrarily large solutions in short time.
Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann
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Norm inflation and low-regularity ill-posedness for the rod equation
The rod equation is ill-posed in H^s(R) for 1 < s < 3/2 by norm inflation from explicit smooth initial data that is arbitrarily small but produces arbitrarily large solutions in short time.