# 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

$\displaystyle u_t + H u_{xx} + u u_x = 0$

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

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

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

## Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

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

where $\displaystyle D_x = \sqrt{-\Delta}$ is the positive differentiation operator. When $\displaystyle a=1$ this is KdV; when $\displaystyle a=0$ this is Benjamin-Ono. Both of these two extreme cases are completely integrable, though the intermediate cases $\displaystyle 0 < a < 1$ are not.

When $\displaystyle 0 < a < 1,$ scaling is $\displaystyle s = -1/2 - a,$ and the following results are known:

• LWP in $\displaystyle H^s$ is known for $\displaystyle s > 9/8 - 3a/8$ KnKoe2003
• For $\displaystyle s \ge 3/4 (2-a)$ this is in KnPoVe1994b
• GWP is known when $\displaystyle s \ge (a+1)/2$ when $\displaystyle a > 4/5,$ 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 $\displaystyle H^s$ norm $\displaystyle || f ||_{H^s}$ with a weighted Sobolev space, namely $\displaystyle || xf ||_{H^{s - 2s_*}},$ where $\displaystyle s_* = (a+1)/2$ is the energy regularity. CoKnSt2003

## Benjamin-Ono with power nonlinearity

This is the equation

$\displaystyle u_t + H u_{xx} + (u^k)_x = 0.$

Thus the original Benjamin-Ono equation corresponds to the case $\displaystyle k=2.$ The scaling exponent is $\displaystyle 1/2 - 1/(k-1).$

• For $\displaystyle k=3,$ one has GWP for large data in $\displaystyle H^1$ KnKoe2003 and LWP for small data in $\displaystyle H^s,$ $\displaystyle s > 1/2$ MlRi2004
• For small data in $\displaystyle H^s,$ $\displaystyle s>1,$ LWP was obtained in KnPoVe1994b
• With the addition of a small viscosity term, GWP can also be obtained in $\displaystyle H^1$ by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in $\displaystyle L^1.$
• For $\displaystyle s < 1/2,$ the solution map is not $\displaystyle C^3$ MlRi2004
• For $\displaystyle k=4,$ LWP for small data in $\displaystyle H^s,$ $\displaystyle s > 5/6$ was obtained in KnPoVe1994b.
• For $\displaystyle k>4,$ LWP for small data in $\displaystyle H^s,$ $\displaystyle s \ge 3/4$ was obtained in KnPoVe1994b.
• For any ${\displaystyle k\geq 3}$ and ${\displaystyle s<1/2-1/k}$ the solution map is not uniformly continuous BiLi2001

## 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 ${\displaystyle L^{2}}$ Li1999, and locally well-posed in ${\displaystyle H^{-3/4+}}$ KozOgTns2001 (see also HuoGuo2005 where ${\displaystyle H^{-1/8+}}$ 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 ${\displaystyle H^{1/4+}}$ HuoGuo2005. For general gKdV-gBO equations one has local well-posedness in ${\displaystyle H^{3}}$ and above GuoTan1992. One can also add damping terms ${\displaystyle Hu_{x}}$ to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.