# 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 [[ | + | ** For s > 9/8 this is in [[KnKoe2003]] |

− | ** For s > 5/4 this is in [[ | + | ** 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 [[ | + | ** For no value of s is the solution map uniformly continuous [[KocTz2005]] |

− | *** For s < -1/2 this is in [[ | + | *** 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 [[ | + | * 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. [[ | + | ** 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 [[ | + | * 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 [[ | + | ** 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 [[ | + | * 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 [[ | + | 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+} [[ | + | 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:

- LWP in H^s for s >= 1 Ta2004
- Global weak solutions exist for L^2 data Sau1979, GiVl1989b, GiVl1991, Tom1990
- Global well-posedness in H^s for s >= 1 Ta2004

## 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
- 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
- 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.