Difference between revisions of "Benjamin-Ono equation"

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The Benjamin-Ono equation (BO) [[Bj1967]], [[On1975]], which models one-dimensional internal waves in deep water, is given by
 
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>
 
<center><math>u_t + H u_{xx} + u u_x = 0</math></center>
where ''H'' is the Hilbert transform. This equation is [[completely integrable]] (see e.g. [[AbFs1983]], [[CoiWic1990]]).
+
where <math>H</math> 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:
+
Scaling is <math>s = -1/2,</math> and the following results are known:
  
* LWP in H^s for s >= 1 [[Ta2004]]
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* LWP in <math>H^s</math> for <math>s \ge 1</math> [[Ta2004]]
** For s > 9/8 this is in [[KnKoe2003]]
+
** For <math>s > 9/8</math> this is in [[KnKoe2003]]
** For s > 5/4 this is in [[KocTz2003]]
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** For <math>s > 5/4</math> this is in [[KocTz2003]]
** For s >= 3/2 this is in [[Po1991]]
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** For <math>s \ge 3/2</math> this is in [[Po1991]]
** For s > 3/2 this is in [[Io1986]]
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** For <math>s > 3/2</math> this is in [[Io1986]]
** For s > 3 this is in [[Sau1979]]
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** For <math>s > 3</math> this is in [[Sau1979]]
 
** For no value of s is the solution map uniformly continuous [[KocTz2005]]
 
** For no value of s is the solution map uniformly continuous [[KocTz2005]]
*** For s < -1/2 this is in [[BiLi2001]]
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*** For <math>s < -1/2</math> this is in [[BiLi2001]]
* Global weak solutions exist for L^2 data [[Sau1979]], [[GiVl1989b]], [[GiVl1991]], [[Tom1990]]
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* 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">H^s</span> for s >= 1 [[Ta2004]]
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* Global well-<span class="SpellE">posedness</span> in <span class="SpellE"><math>H^s</math></span> for <math>s \ge 1</math> [[Ta2004]]
** For s >= 3/2 this is in [[Po1991]]
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** For <math>s \ge 3/2</math> this is in [[Po1991]]
 
** For smooth solutions this is in [[Sau1979]]
 
** For smooth solutions this is in [[Sau1979]]
  
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<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>
  
where <math>D_x := \sqrt{-\Delta}</math> 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.
+
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 0 < a < 1, scaling is s = -1/2 - a, and the following results are known:
+
When <math>0 < a < 1,</math> scaling is <math>s = -1/2 - a,</math> and the following results are known:
  
* LWP in H^s is known for s > 9/8 - 3a/8 [[KnKoe2003]]
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* LWP in <math>H^s</math> is known for <math>s > 9/8 - 3a/8</math> [[KnKoe2003]]
** For s >= 3/4 (2-a) this is in [[KnPoVe1994b]]
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** For <math>s \ge 3/4 (2-a)</math> this is in [[KnPoVe1994b]]
* GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [[KnPoVe1994b]]
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* 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]]
 
* 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]]
+
** 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 ==
 
== Benjamin-Ono with power nonlinearity ==
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This is the equation
 
This is the equation
 
<center><math> u_t + H u_{xx} + (u^k)_x = 0.</math></center>
 
<center><math> u_t + H u_{xx} + (u^k)_x = 0.</math></center>
Thus the original Benjamin-Ono equation corresponds to the case ''k=2''.
+
Thus the original Benjamin-Ono equation corresponds to the case <math>k=2.</math>
The scaling exponent is 1/2 - 1/(k-1).
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The scaling exponent is <math>1/2 - 1/(k-1).</math>
  
* For k=3, one has GWP for large data in H^1 [[KnKoe2003]] and LWP for small data in H^s, s > ½ [[MlRi2004]]
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* 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 H^s, s>1, LWP was obtained in [[KnPoVe1994b]]
+
** 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 H^1 by complete integrability methods in [[FsLu2000]], with asymptotics under the additional assumption that the initial data is in L^1.
+
** 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 s < ½, the solution map is not C^3 [[MlRi2004]]
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** For <math>s < 1/2,</math> the solution map is not <math>C^3</math> [[MlRi2004]]
* For k=4, LWP for small data in H^s, s > 5/6 was obtained in [[KnPoVe1994b]].
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* For <math>k=4,</math> LWP for small data in <math>H^s,</math> <math>s > 5/6</math> was obtained in [[KnPoVe1994b]].
* For k>4, LWP for small data in H^s, s >=3/4 was obtained in [[KnPoVe1994b]].
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* 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 k >= 3 and s < 1/2 - 1/k the solution map is not uniformly continuous [[BiLi2001]]
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* 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 ==
 
== 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).  
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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 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 Hu_x to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping [[OttSud1970]].
+
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]]

Revision as of 16:21, 5 May 2007

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.