Benjamin-Ono equation: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
No edit summary
No edit summary
Line 7: Line 7:
Scaling is s = -1/2, and the following results are known:
Scaling is s = -1/2, and the following results are known:


* LWP in <span class="SpellE">H^s</span> 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 [[KnKoe-p]]
** For s >= 5/4 this is in [[KocTz-p]]
** For s >= 5/4 this is in [[KocTz-p]]
Line 20: Line 20:
** For smooth solutions this is in [[Sau1979]]
** For smooth solutions this is in [[Sau1979]]


== Generalised Benjamin-Ono equation ==
== Generalized Benjamin-Ono equation ==


The ''generalized Benjamin-Ono equation''  is the scalar equation
The ''generalized Benjamin-Ono equation''  is the scalar equation
Line 27: Line 27:
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 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 - <span class="GramE">a,</span> and the following results are known:
When 0 < a < 1, scaling is s = -1/2 - a, and the following results are known:


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


== Higher order Benjamin-Ono ==
== Benjamin-Ono with power nonlinearity ==


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).
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 ''k=2''.
The scaling exponent is 1/2 - 1/(k-1).


* 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 [[MlRi-p]]
* 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 <span class="SpellE">H^s</span>, 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 <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 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 [[MlRi-p]]
* For k=4, LWP for small data in <span class="SpellE">H^s</span>, 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 <span class="SpellE">H^s</span>, 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 [[BiLi-p]]


== Other generalizations ==
== Other generalizations ==


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 [[Li1999]], and locally well-posed in H<span class="GramE">^{</span>-3/4+} [[KozOgTns2001]] (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 <span class="SpellE">KdV</span>-BO equation, which is locally well-posed in H<span class="GramE">^{</span>1/4+} [[HuoGuo-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 [[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 [[OttSud1982]].
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).  
 
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 [[OttSud1982]].


[[Category:Integrability]]
[[Category:Integrability]]
[[Category:Equations]]
[[Category:Equations]]

Revision as of 20:50, 11 August 2006

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 KnKoe-p
  • 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

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 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

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).

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 OttSud1982.