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<center>'''Benjamin-Ono equation'''</center>
<center>'''Benjamin-Ono equation'''</center>


[Thanks to <span class="SpellE">Nikolay</span> <span class="SpellE">Tzvetkov</span> and Felipe <span class="SpellE">Linares</span> for help with this section - Ed]
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>
where <math>H</math> is the [[Hilbert transform]]. This equation is [[completely integrable]] (see e.g., [[AbFs1983]], [[CoiWic1990]]).


The ''generalized Benjamin-Ono equation'' <span class="SpellE">BO_a</span> is the scalar equation
Scaling is <math>s = -1/2,</math> and the following results are known:


<center><span class="SpellE">u_t</span> + <span class="SpellE">D_x</span><span class="GramE">^{</span>1+a} <span class="SpellE">u_x</span> + <span class="SpellE">uu_x</span> = 0</center>
* LWP in <math>H^s</math> for <math>s \ge 1</math> [[Ta2004]]
** For <math>s > 9/8</math> this is in [[KnKoe2003]]
** For <math>s > 5/4</math> this is in [[KocTz2003]]
** For <math>s \ge 3/2</math> this is in [[Po1991]]
** For <math>s > 3/2</math> this is in [[Io1986]]
** For <math>s > 3</math> this is in [[Sau1979]]
** For no value of s is the solution map uniformly continuous [[KocTz2005]]
*** For <math>s < -1/2</math> this is in [[BiLi2001]]
* 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"><math>H^s</math></span> for <math>s \ge 1</math> [[Ta2004]]
** For <math>s \ge 3/2</math> this is in [[Po1991]]
** For smooth solutions this is in [[Sau1979]]


<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>[references.html#Bj1967 Bj1967]], [[references.html#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. [[references.html#AbFs1983 AbFs1983]], [[references.html#CoiWic1990 CoiWic1990]]), though the intermediate cases 0 < a < 1 are not.
== Generalized Benjamin-Ono equation ==


When a=0, scaling is s = -1/2, and the following results are known:
The ''generalized Benjamin-Ono equation''  is the scalar equation
<center><math>\partial_t u  + D_x^{1+a} \partial_x u + u\partial_x u = 0.</math></center>


* LWP in <span class="SpellE">H^s</span> for s >= 1 [Ta-p]
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.
** 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 >= 3/2 this is in [[references.html#Po1991 Po1991]]
** For s > 3/2 this is in [[references.html#Io1986 Io1986]]
** For s > 3 this is in [[references.html#Sau1979 Sau1979]]
** 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]
* Global weak solutions exist for L^2 data [[references.html#Sau1979 Sau1979]], [[references.html#GiVl1989b GiVl1989b]], [[references.html#GiVl1991 GiVl1991]], [[references.html#Tom1990 Tom1990]]
* 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 [[references.html#Po1991 Po1991]]
** For smooth solutions this is in [[references.html#Sau1979 Sau1979]]


When 0 < a < 1, scaling is s = -1/2 - <span class="GramE">a,</span> 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 <span class="SpellE">H^s</span> is known for s > 9/8 – 3a/8 [<span class="SpellE">KnKoe</span>-p]
* 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 [[references.html#KnPoVe1994b KnPoVe1994b]]
** 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 [[references.html#KnPoVe1994b KnPoVe1994b]]
* 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 [[references.html#MlSauTz2001 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 <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]]


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).
== Benjamin-Ono with power nonlinearity ==


* 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]
This is the equation
** For small data in <span class="SpellE">H^s</span>, s>1, LWP was obtained in [[references.html#KnPoVe1994b KnPoVe1994b]]
<center><math> u_t + H u_{xx} + (u^k)_x = 0.</math></center>
** 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.
Thus the original Benjamin-Ono equation corresponds to the case <math>k=2.</math>
** For s < ½, the solution map is not C^3 [<span class="SpellE">MlRi</span>-p]
The scaling exponent is <math>1/2 - 1/(k-1).</math>
* For k=4, LWP for small data in <span class="SpellE">H^s</span>, s > 5/6 was obtained in [[references.html#KnPoVe1994b KnPoVe1994b]].
* For k>4, LWP for small data in <span class="SpellE">H^s</span>, s >=3/4 was obtained in [[references.html#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]


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 [[references.html#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 [[references.html#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 [[references.html#OttSud1982 OttSud1982]].
* 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 <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 <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 <math>s < 1/2,</math> the solution map is not <math>C^3</math> [[MlRi2004]]
* For <math>k=4,</math> LWP for small data in <math>H^s,</math> <math>s > 5/6</math> was obtained in [[KnPoVe1994b]].
* 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 <math>k \ge 3</math> and <math>s < 1/2 - 1/k</math> 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 <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 <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:Equations]]
[[Category:Equations]]

Latest revision as of 18:49, 25 October 2008

Benjamin-Ono equation

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H} is the Hilbert transform. This equation is completely integrable (see e.g., AbFs1983, CoiWic1990).

Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s = -1/2,} and the following results are known:

  • LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1} Ta2004
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 9/8} this is in KnKoe2003
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 5/4} this is in KocTz2003
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 3/2} this is in Po1991
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 3/2} this is in Io1986
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 3} this is in Sau1979
    • For no value of s is the solution map uniformly continuous KocTz2005
      • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s < -1/2} this is in BiLi2001
  • Global weak solutions exist for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2} data Sau1979, GiVl1989b, GiVl1991, Tom1990
  • Global well-posedness in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1} Ta2004
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 3/2} this is in Po1991
    • For smooth solutions this is in Sau1979

Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u + D_x^{1+a} \partial_x u + u\partial_x u = 0.}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_x = \sqrt{-\Delta}} is the positive differentiation operator. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a=1} this is KdV; when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a=0} this is Benjamin-Ono. Both of these two extreme cases are completely integrable, though the intermediate cases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 < a < 1} are not.

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 < a < 1,} scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s = -1/2 - a,} and the following results are known:

  • LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 9/8 - 3a/8} KnKoe2003
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 3/4 (2-a)} this is in KnPoVe1994b
  • GWP is known when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge (a+1)/2} when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} norm Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle || f ||_{H^s}} with a weighted Sobolev space, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle || xf ||_{H^{s - 2s_*}},} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_* = (a+1)/2} is the energy regularity. CoKnSt2003

Benjamin-Ono with power nonlinearity

This is the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_t + H u_{xx} + (u^k)_x = 0.}

Thus the original Benjamin-Ono equation corresponds to the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=2.} The scaling exponent is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/2 - 1/(k-1).}

  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=3,} one has GWP for large data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} KnKoe2003 and LWP for small data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1/2} MlRi2004
    • For small data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>1,} LWP was obtained in KnPoVe1994b
    • With the addition of a small viscosity term, GWP can also be obtained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^1.}
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s < 1/2,} the solution map is not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^3} MlRi2004
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=4,} LWP for small data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 5/6} was obtained in KnPoVe1994b.
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k>4,} LWP for small data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 3/4} 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.

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