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


[Thanks to  and Felipe Linares for help with this section - Ed]
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
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>
<center><math>\partial_t u  + D_x^{1+a} \partial_x u + u\partial_x u = 0.</math></center>


<span class="GramE">where</span> <span class="SpellE">D_x</span> = <span class="SpellE">sqrt</span>{-Delta} is the positive differentiation operator. When a=1 this is [#kdv <span class="SpellE">KdV</span>]<nowiki>; when a=0 this is the Benjamin-Ono equation (BO) [</nowiki>[Bibliography#Bj1967|Bj1967]], [[Bibliography#On1975|On1975]], which models one-dimensional internal waves in deep water. Both of these equations are completely <span class="SpellE">integrable</span> (see e.g. [[Bibliography#AbFs1983|AbFs1983]], [[Bibliography#CoiWic1990|CoiWic1990]]), though the intermediate cases 0 < a < 1 are not.
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:


When a=0, scaling is s = -1/2, 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]]


* LWP in <span class="SpellE">H^s</span> for s >= 1 [Ta-p]
== Benjamin-Ono with power nonlinearity ==
** For s >= 9/8 this is in [<span class="SpellE">KnKoe</span>-p]
** For s >= 5/4 this is in [<span class="SpellE">KocTz</span>-p]
** For s >= 3/2 this is in [[Bibliography#Po1991|Po1991]]
** For s > 3/2 this is in [[Bibliography#Io1986|Io1986]]
** For s > 3 this is in [[Bibliography#Sau1979|Sau1979]]
** For no value of s is the solution map uniformly continuous [KocTz-p2]
*** For s < -1/2 this is in [<span class="SpellE">BiLi</span>-p]
* Global weak solutions exist for L^2 data [[Bibliography#Sau1979|Sau1979]], [[Bibliography#GiVl1989b|GiVl1989b]], [[Bibliography#GiVl1991|GiVl1991]], [[Bibliography#Tom1990|Tom1990]]
* Global well-<span class="SpellE">posedness</span> in <span class="SpellE">H^s</span> for s >= 1 [Ta-p]
** For s >= 3/2 this is in [[Bibliography#Po1991|Po1991]]
** For smooth solutions this is in [[Bibliography#Sau1979|Sau1979]]


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


* LWP in <span class="SpellE">H^s</span> is known for s > 9/8 \u2013 3a/8 [<span class="SpellE">KnKoe</span>-p]
* 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 s >= 3/4 (2-a) this is in [[Bibliography#KnPoVe1994b|KnPoVe1994b]]
** For small data in <math>H^s,</math> <math>s>1,</math> LWP was obtained in [[KnPoVe1994b]]
* GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [[Bibliography#KnPoVe1994b|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>
* The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work [[Bibliography#MlSauTz2001|MlSauTz2001]]
** For <math>s < 1/2,</math> the solution map is not <math>C^3</math> [[MlRi2004]]
** However, this can be salvaged by combining the <span class="SpellE">H^s</span> norm || f ||_{<span class="SpellE">H^s</span>} with a weighted <span class="SpellE">Sobolev</span> space, namely || <span class="SpellE">xf</span> ||_{H^{s - 2s_*}}, where s_* = (a+1)/2 is the energy regularity.[CoKnSt-p4]
* 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]]


One can replace the quadratic non-linearity <span class="SpellE">uu_x</span> by higher powers u<span class="GramE">^{</span>k-1} <span class="SpellE">u_x</span>, in analogy with <span class="SpellE">KdV</span> and <span class="SpellE">gKdV</span>, giving rise to the <span class="SpellE">gBO</span>-k equations (let us take a=0 for sake of discussion).The scaling exponent is 1/2 - 1<span class="GramE">/(</span>k-1).
== Other generalizations ==


* For k=3, one has GWP for large data in H^1 [<span class="SpellE">KnKoe</span>-p] and LWP for small data in <span class="SpellE">H^s</span>, s > ½ [<span class="SpellE">MlRi</span>-p]
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).
** For small data in <span class="SpellE">H^s</span>, s>1, LWP was obtained in [[Bibliography#KnPoVe1994b|KnPoVe1994b]]
** With the addition of a small viscosity term, GWP can also be obtained in H^1 by complete <span class="SpellE">integrability</span> methods in [FsLu2000], with <span class="SpellE">asymptotics</span> under the additional assumption that the initial data is in L^1.
** For s < ½, the solution map is not C^3 [<span class="SpellE">MlRi</span>-p]
* For k=4, LWP for small data in <span class="SpellE">H^s</span>, s > 5/6 was obtained in [[Bibliography#KnPoVe1994b|KnPoVe1994b]].
* For k>4, LWP for small data in <span class="SpellE">H^s</span>, s >=3/4 was obtained in [[Bibliography#KnPoVe1994b|KnPoVe1994b]].
* For any k >= 3 and s < 1/2 - 1/k the solution map is not uniformly continuous [<span class="SpellE">BiLi</span>-p]


The <span class="SpellE">KdV</span>-Benjamin Ono (<span class="SpellE">KdV</span>-BO) equation is formed by combining the linear parts of the <span class="SpellE">KdV</span> and Benjamin-Ono equations together.It is globally well-posed in L^2 [[Bibliography#Li1999|Li1999]], and locally well-posed in H<span class="GramE">^{</span>-3/4+} [<span class="SpellE">KozOgTns</span>] (see also [<span class="SpellE">HuoGuo</span>-p] where H^{-1/8+} is obtained).Similarly one can generalize the non-linearity to be k-linear, generating for instance the modified <span class="SpellE">KdV</span>-BO equation, which is locally well-posed in H<span class="GramE">^{</span>1/4+} [<span class="SpellE">HuoGuo</span>-p].For general <span class="SpellE">gKdV-gBO</span> equations one has local well-<span class="SpellE"><span class="GramE">posedness</span></span><span class="GramE">in</span> H^3 and above [[Bibliography#GuoTan1992|GuoTan1992]].One can also add damping terms <span class="SpellE">Hu_x</span> to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping [[Bibliography#OttSud1982|OttSud1982]].
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:Integrability]]
[[Category:Equations]]
[[Category:Equations]]

Latest revision as of 18:49, 25 October 2008

Benjamin-Ono equation

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

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

Scaling is and the following results are known:

  • LWP in for Ta2004
    • For this is in KnKoe2003
    • For this is in KocTz2003
    • For this is in Po1991
    • For this is in Io1986
    • For this is in Sau1979
    • For no value of s is the solution map uniformly continuous KocTz2005
      • For this is in BiLi2001
  • Global weak solutions exist for data Sau1979, GiVl1989b, GiVl1991, Tom1990
  • Global well-posedness in for Ta2004
    • For this is in Po1991
    • For smooth solutions this is in Sau1979

Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

where is the positive differentiation operator. When this is KdV; when this is Benjamin-Ono. Both of these two extreme cases are completely integrable, though the intermediate cases are not.

When scaling is and the following results are known:

  • LWP in is known for KnKoe2003
    • For this is in KnPoVe1994b
  • GWP is known when when 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 norm with a weighted Sobolev space, namely where is the energy regularity. CoKnSt2003

Benjamin-Ono with power nonlinearity

This is the equation

Thus the original Benjamin-Ono equation corresponds to the case The scaling exponent is

  • For one has GWP for large data in KnKoe2003 and LWP for small data in MlRi2004
    • For small data in LWP was obtained in KnPoVe1994b
    • With the addition of a small viscosity term, GWP can also be obtained in by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in
    • For the solution map is not MlRi2004
  • For LWP for small data in was obtained in KnPoVe1994b.
  • For LWP for small data in was obtained in KnPoVe1994b.
  • For any and 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 Li1999, and locally well-posed in KozOgTns2001 (see also HuoGuo2005 where 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 HuoGuo2005. For general gKdV-gBO equations one has local well-posedness in and above GuoTan1992. One can also add damping terms to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.