# Difference between revisions of "Benjamin-Ono equation"

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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> | ||

− | <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>[ | + | <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. |

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

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** For s >= 9/8 this is in [<span class="SpellE">KnKoe</span>-p] | ** 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 >= 5/4 this is in [<span class="SpellE">KocTz</span>-p] | ||

− | ** For s >= 3/2 this is in [[ | + | ** For s >= 3/2 this is in [[Bibliography#Po1991|Po1991]] |

− | ** For s > 3/2 this is in [[ | + | ** For s > 3/2 this is in [[Bibliography#Io1986|Io1986]] |

− | ** For s > 3 this is in [[ | + | ** For s > 3 this is in [[Bibliography#Sau1979|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 [KocTz-p2] | ||

*** For s < -1/2 this is in [<span class="SpellE">BiLi</span>-p] | *** For s < -1/2 this is in [<span class="SpellE">BiLi</span>-p] | ||

− | * Global weak solutions exist for L^2 data [[ | + | * 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] | * 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 [[ | + | ** For s >= 3/2 this is in [[Bibliography#Po1991|Po1991]] |

− | ** For smooth solutions this is in [[ | + | ** 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: | When 0 < a < 1, scaling is s = -1/2 - <span class="GramE">a,</span> and the following results are known: | ||

− | * LWP in <span class="SpellE">H^s</span> is known for s > 9/8 | + | * LWP in <span class="SpellE">H^s</span> is known for s > 9/8 \u2013 3a/8 [<span class="SpellE">KnKoe</span>-p] |

− | ** For s >= 3/4 (2-a) this is in [[ | + | ** For s >= 3/4 (2-a) this is in [[Bibliography#KnPoVe1994b|KnPoVe1994b]] |

− | * GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [[ | + | * GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [[Bibliography#KnPoVe1994b|KnPoVe1994b]] |

− | * The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work [[ | + | * The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work [[Bibliography#MlSauTz2001|MlSauTz2001]] |

** 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] | ** 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] | ||

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* 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] | * 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] | ||

− | ** For small data in <span class="SpellE">H^s</span>, s>1, LWP was obtained in [[ | + | ** 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. | ** 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 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 [[ | + | * 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 [[ | + | * 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] | * 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 [[ | + | 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]]. |

[[Category:Equations]] | [[Category:Equations]] |

## Revision as of 19:43, 28 July 2006

**Benjamin-Ono equation**

[Thanks to and Felipe Linares for help with this section - Ed]

The *generalized Benjamin-Ono equation* is the scalar equation

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

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

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

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 \u2013 3a/8 [KnKoe-p]
- 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.[CoKnSt-p4]

One can replace the quadratic non-linearity uu_x by higher powers u^{k-1} u_x, in analogy with KdV and gKdV, giving rise to the gBO-k equations (let us take a=0 for sake of discussion).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 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 [MlRi-p]

- 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 [BiLi-p]

The KdV-Benjamin Ono (KdV-BO) 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+} [KozOgTns] (see also [HuoGuo-p] 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-posednessin 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 OttSud1982.