Scaling is s = − 1 / 2, and the following results are known:
- LWP in Hs for Ta2004
- Global weak solutions exist for L2 data Sau1979, GiVl1989b, GiVl1991, Tom1990
- Global well-posedness in Hs for Ta2004
Generalized Benjamin-Ono equation
The generalized Benjamin-Ono equation is the scalar equation
where is the positive differentiation operator. When a = 1 this is KdV; when a = 0 this is Benjamin-Ono. Both of these two extreme cases are completely integrable, though the intermediate cases 0 < a < 1 are not.
When 0 < a < 1, scaling is s = − 1 / 2 − a, and the following results are known:
- LWP in Hs is known for s > 9 / 8 − 3a / 8 KnKoe2003
- For this is in KnPoVe1994b
- GWP is known when when a > 4 / 5, from the conservation of the Hamiltonian KnPoVe1994b
- The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work MlSauTz2001
- However, this can be salvaged by combining the Hs norm with a weighted Sobolev space, namely where s * = (a + 1) / 2 is the energy regularity. CoKnSt2003
Benjamin-Ono with power nonlinearity
This is the equation
Thus the original Benjamin-Ono equation corresponds to the case k = 2. The scaling exponent is 1 / 2 − 1 / (k − 1).
- For k = 3, one has GWP for large data in H1 KnKoe2003 and LWP for small data in Hs, s > 1 / 2 MlRi2004
- For small data in Hs, s > 1, LWP was obtained in KnPoVe1994b
- With the addition of a small viscosity term, GWP can also be obtained in H1 by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in L1.
- For s < 1 / 2, the solution map is not C3 MlRi2004
- For k = 4, LWP for small data in Hs, s > 5 / 6 was obtained in KnPoVe1994b.
- For k > 4, LWP for small data in Hs, was obtained in KnPoVe1994b.
- For any and s < 1 / 2 − 1 / k the solution map is not uniformly continuous BiLi2001
The KdV-Benjamin Ono equation is formed by combining the linear parts of the KdV and Benjamin-Ono equations together. It is globally well-posed in L2 Li1999, and locally well-posed in H − 3 / 4 + KozOgTns2001 (see also HuoGuo2005 where H − 1 / 8 + is obtained).
Similarly one can generalize the non-linearity to be k-linear, generating for instance the modified KdV-BO equation, which is locally well-posed in H1 / 4 + HuoGuo2005. For general gKdV-gBO equations one has local well-posedness in H3 and above GuoTan1992. One can also add damping terms Hux to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.