Benjamin-Ono equation

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Benjamin-Ono equation

The Benjamin-Ono equation (BO) Bj1967, On1975, which models one-dimensional internal waves in deep water, is given by

u t + H u x x + u u x = 0

where $H$ is the Hilbert transform. This equation is completely integrable (see e.g., AbFs1983, CoiWic1990).

Scaling is $s = - 1 / 2,$ and the following results are known:

LWP in H^s for Ta2004
- For $s > 9 / 8$ this is in KnKoe2003
- For $s > 5 / 4$ this is in KocTz2003
- For $s \ge 3/2$ this is in Po1991
- For $s > 3 / 2$ this is in Io1986
- For $s > 3$ this is in Sau1979
- For no value of s is the solution map uniformly continuous KocTz2005
  - For $s < - 1 / 2$ this is in BiLi2001
Global weak solutions exist for $L 2$ data Sau1979, GiVl1989b, GiVl1991, Tom1990
Global well-posedness in $H s$ for Ta2004
- For $s \ge 3/2$ this is in Po1991
- For smooth solutions this is in Sau1979

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Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

$\partial_t u + D_x^{1+a} \partial_x u + u\partial_x u = 0.$

where $D_x = \sqrt{-\Delta}$ 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 H^s is known for s > 9 / 8 − 3a / 8 KnKoe2003
- For $s \ge 3/4 (2-a)$ this is in KnPoVe1994b
GWP is known when $s \ge (a+1)/2$ 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 $H s$ norm $|| f ||_{H^s}$ with a weighted Sobolev space, namely $|| xf ||_{H^{s - 2s_*}},$ where $s * = (a + 1) / 2$ is the energy regularity. CoKnSt2003

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Benjamin-Ono with power nonlinearity

This is the equation

u t + H u x x + (u k) x = 0.

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 H^s, s > 1 / 2 MlRi2004
- For small data in $H s,$ $s > 1,$ LWP was obtained in KnPoVe1994b
- With the addition of a small viscosity term, GWP can also be obtained in $H 1$ by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in $L 1 .$
- For $s < 1 / 2,$ the solution map is not $C 3$ MlRi2004
For $k = 4,$ LWP for small data in $H s,$ $s > 5 / 6$ was obtained in KnPoVe1994b.
For $k > 4,$ LWP for small data in $H s,$ $s \ge 3/4$ was obtained in KnPoVe1994b.
For any $k \ge 3$ and $s < 1 / 2 - 1 / k$ the solution map is not uniformly continuous BiLi2001

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Other generalizations

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 $L 2$ 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 $H 1 / 4 +$ HuoGuo2005. For general gKdV-gBO equations one has local well-posedness in $H 3$ and above GuoTan1992. One can also add damping terms $H u x$ to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.