View source for Benjamin-Ono equation

<center>'''Benjamin-Ono equation'''</center>

The Benjamin-Ono equation (BO) [[Bj1967]], [[On1975]], which models one-dimensional internal waves in deep water, is given by
<center><math>u_t + H u_{xx} + u u_x = 0</math></center>
where <math>H</math> is the [[Hilbert transform]]. This equation is [[completely integrable]] (see e.g., [[AbFs1983]], [[CoiWic1990]]).

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

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

== Generalized Benjamin-Ono equation ==

The ''generalized Benjamin-Ono equation''  is the scalar equation
<center><math>\partial_t u  + D_x^{1+a} \partial_x u + u\partial_x u = 0.</math></center>

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

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

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

== Benjamin-Ono with power nonlinearity ==

This is the equation
<center><math> u_t + H u_{xx} + (u^k)_x = 0.</math></center>
Thus the original Benjamin-Ono equation corresponds to the case <math>k=2.</math>
The scaling exponent is <math>1/2 - 1/(k-1).</math>

* For <math>k=3,</math> one has GWP for large data in <math>H^1</math> [[KnKoe2003]] and LWP for small data in <math>H^s,</math> <math>s > 1/2</math> [[MlRi2004]]
** For small data in <math>H^s,</math> <math>s>1,</math> LWP was obtained in [[KnPoVe1994b]]
** With the addition of a small viscosity term, GWP can also be obtained in <math>H^1</math> by complete integrability methods in [[FsLu2000]], with asymptotics under the additional assumption that the initial data is in <math>L^1.</math>
** For <math>s < 1/2,</math> the solution map is not <math>C^3</math> [[MlRi2004]]
* For <math>k=4,</math> LWP for small data in <math>H^s,</math> <math>s > 5/6</math> was obtained in [[KnPoVe1994b]].
* For <math>k>4,</math> LWP for small data in <math>H^s,</math> <math>s \ge 3/4</math> was obtained in [[KnPoVe1994b]].
* For any <math>k \ge 3</math> and <math>s < 1/2 - 1/k</math> 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 <math>L^2</math> [[Li1999]], and locally well-posed in <math>H^{-3/4+}</math> [[KozOgTns2001]] (see also [[HuoGuo2005]] where <math>H^{-1/8+}</math> 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 <math>H^{1/4+}</math> [[HuoGuo2005]]. For general gKdV-gBO equations one has local well-posedness in <math>H^3</math> and above [[GuoTan1992]]. One can also add damping terms <math>Hu_x</math> to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping [[OttSud1970]].

[[Category:Integrability]]
[[Category:Equations]]