Difference between revisions of "Benjamin-Ono equation"

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* LWP in H^s for s >= 1 [[Ta2004]]
 
* LWP in H^s for s >= 1 [[Ta2004]]
** For s > 9/8 this is in [[KnKoe-p]]
+
** For s > 9/8 this is in [[KnKoe2003]]
** For s > 5/4 this is in [[KocTz-p]]
+
** For s > 5/4 this is in [[KocTz2003]]
 
** For s >= 3/2 this is in [[Po1991]]
 
** For s >= 3/2 this is in [[Po1991]]
 
** For s > 3/2 this is in [[Io1986]]
 
** For s > 3/2 this is in [[Io1986]]
 
** For s > 3 this is in [[Sau1979]]
 
** For s > 3 this is in [[Sau1979]]
** For no value of s is the solution map uniformly continuous [[KocTz-p2]]
+
** For no value of s is the solution map uniformly continuous [[KocTz2005]]
*** For s < -1/2 this is in [[BiLi-p]]
+
*** For s < -1/2 this is in [[BiLi2001]]
 
* Global weak solutions exist for L^2 data [[Sau1979]], [[GiVl1989b]], [[GiVl1991]], [[Tom1990]]
 
* Global weak solutions exist for L^2 data [[Sau1979]], [[GiVl1989b]], [[GiVl1991]], [[Tom1990]]
 
* Global well-<span class="SpellE">posedness</span> in <span class="SpellE">H^s</span> for s >= 1 [[Ta2004]]
 
* Global well-<span class="SpellE">posedness</span> in <span class="SpellE">H^s</span> for s >= 1 [[Ta2004]]
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When 0 < a < 1, scaling is s = -1/2 - a, and the following results are known:
 
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 [[KnKoe-p]]
+
* LWP in H^s is known for s > 9/8 - 3a/8 [[KnKoe2003]]
 
** For s >= 3/4 (2-a) this is in [[KnPoVe1994b]]
 
** For s >= 3/4 (2-a) this is in [[KnPoVe1994b]]
 
* GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [[KnPoVe1994b]]
 
* GWP is known when s >= (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]]
 
* 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. [[CoKnSt-p4]]
+
** 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]]
  
 
== Benjamin-Ono with power nonlinearity ==
 
== Benjamin-Ono with power nonlinearity ==
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The scaling exponent is 1/2 - 1/(k-1).
 
The scaling exponent is 1/2 - 1/(k-1).
  
* For k=3, one has GWP for large data in H^1 [[KnKoe-p]] and LWP for small data in H^s, s > ½ [[MlRi-p]]
+
* For k=3, one has GWP for large data in H^1 [[KnKoe2003]] and LWP for small data in H^s, s > ½ [[MlRi2004]]
 
** For small data in H^s, s>1, LWP was obtained in [[KnPoVe1994b]]
 
** 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.
 
** 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 < ½, the solution map is not C^3 [[MlRi-p]]
+
** For s < ½, 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 > 5/6 was obtained in [[KnPoVe1994b]].
 
* For k>4, LWP for small data in H^s, s >=3/4 was obtained in [[KnPoVe1994b]].
 
* For k>4, LWP for small data in H^s, s >=3/4 was obtained in [[KnPoVe1994b]].
* For any k >= 3 and s < 1/2 - 1/k the solution map is not uniformly continuous [[BiLi-p]]
+
* For any k >= 3 and s < 1/2 - 1/k 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 [[HuoGuo-p]] where H^{-1/8+} is obtained).  
+
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+} [[HuoGuo-p]]. 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 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]].
  
 
[[Category:Integrability]]
 
[[Category:Integrability]]
 
[[Category:Equations]]
 
[[Category:Equations]]

Revision as of 00:59, 17 March 2007

Benjamin-Ono equation

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

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:

Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

where 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
  • GWP is known when s >= (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

Benjamin-Ono with power nonlinearity

This is the equation

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 H^1 KnKoe2003 and LWP for small data in H^s, s > ½ 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 < ½, 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 >=3/4 was obtained in KnPoVe1994b.
  • For any k >= 3 and 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 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 Hu_x to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.