Bilinear Airy estimates

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Algebraic identity

Much of the bilinear estimate theory for Airy equation rests on the following ``three-wave resonance identity:

Estimates

The following bilinear estimates are known:

|| (uv)_x ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}

|| (uv)_x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}
|| (uv)_x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}
  • Remark: In principle, a complete list of bilinear estimates could be obtained from [[references.html#Ta-p2 Ta-p2]].