Surface quasi-geostrophic equation and Template:Reflist: Difference between pages

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The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \R$. In the inviscid case the equation is $$ \theta_t + u \cdot \nabla \theta = 0,$$ where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
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Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$
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The equation is used as a toy model for the 3D Euler equation and Navier-Stokes. The main question is to determine whether the Cauchy problem is well posed in the classical sense. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions <ref name="CMT"/>. The same comparison can be made between the supercritical SQG equation and Navier-Stokes.
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The key feature of the model is that the drift $u$ is a divergence free vector field related to the solution $\theta$ by a zeroth order singular integral operator.
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For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.
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Global weak solutions, as well as classical solutions locally in time, are known to exist globally for the full range of $s \in [0,1]$ <ref name="R" />.
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== Conserved quantities ==
    | #default = decimal}}}}};">
 
{{#tag:references|{{{refs|}}}|group={{{group|}}}}}</div><noinclude>
The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).
{{documentation}}</noinclude>
 
* ''' Maximum principle '''
 
The supremum of $\theta$ occurs at time zero: $||\theta(t,.)||_{L^\infty} \leq ||\theta(0,.)||_{L^\infty}$.
 
* '''Conservation of energy'''.
 
A classical solution $u$ satisfies the energy equality
$$ \int_{\R^2} \theta(0,x)^2 \ dx = \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$
 
In the case of weak solutions, only the energy inequality is available
$$ \int_{\R^2} \theta(0,x)^2 \ dx \geq \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$
 
* '''$H^{-1/2}$ estimate'''
 
The $H^{-1/2}$ norm of $\theta$ does not increase in time.
 
$$ \int_{\R^2} |(-\Delta)^{-1/4} \theta(0,x)|^2 \ dx = \int_{\R^2} |(-\Delta)^{-1/4}\theta(t,x)|^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2-1/4}\theta(r,x)|^2 \ dx \ dr.$$
 
 
== Scaling and criticality ==
 
If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.
 
The $L^\infty$ norm is invariant by the scaling of the equation if $s=1/2$. This observation makes $s=1/2$ the critical exponent for the strongest a priori estimate available. For larger values of $s$, the diffusion dominates the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.
 
== Well posedness results ==
 
=== Sub-critical case: $s>1/2$ ===
 
The equation is well posed globally. The proof can be done with [[perturbation methods]] using only soft functional analysis or Fourier analysis.
 
=== Critical case: $s=1/2$ ===
 
The equation is well posed globally. There are four known proofs.
 
* '''Evolution of a modulus of continuity''' <ref name="KNV"/>: An explicit modulus of continuity which is comparable to Lipschitz in small scales but growth logarithmically in large scales is shown to be preserved by the flow. The method is vaguely comparable to [[Ishii-Lions]].
* '''De Giorgi approach''' <ref name="CV"/>: From the $L^\infty$ modulus of continuity, it is concluded that $u$ stays bounded in $BMO$. A variation to the parabolic [[De Giorgi-Nash-Moser]] can be carried out to obtain Holder continuity of $\theta$. The result does not use the relations $u = R^\perp \theta$, but only that $u$ is a divergence-free vector field in ''BMO''. Therefore, it is actually a regularity result for arbitrary [[drift-diffusion equations]].
* '''Dual flow method''' <ref name="KN"/>: Also from the information that $u$ is $BMO$ and divergence free, it can be shown that the solution $\theta$ becomes Holder continuous by studying the dual flow and characterizing Holder functions in terms of how they integrate against simple test functions. This is a regularity result for general [[drift-diffusion equations]] as well.
* '''Nonlinear maximum principle''' <ref name="ConstVicol"/>: By studying the evolution of $|\nabla \theta|^2$ and using a [[nonlinear lower bound on the fractional Laplacian]] when evaluated at extrema, one may prove that a solution of SQG which has only (sufficiently) small jumps, then it is in fact smooth. By looking at the evolution of finite differences $\delta_h \theta(x) = \theta(x+h) - \theta(x)$, one may measure the evolution of these small jumps, and  prove that if the initial data has only small jumps, then so does the corresponding solution of SQG. Combined, the above statements imply global regularity.
 
=== Supercritical case: $s<1/2$ ===
 
The global well posedness of the equation is an open problem. Some partial results are known:
 
* Existence of solutions locally in time.
* Existence of global weak solutions. <ref name="R"/>
* Global smooth solution if the initial data is sufficiently small. <ref name="Y"/>
* Smoothness of weak solutions for sufficiently large time. <ref name="S"/> <ref name="D"/> <ref name="K"/>
 
=== Inviscid case ===
 
The global well posedness of the equation is an open problem. Some partial results are known:
 
* Existence of smooth solutions locally in time{{Citation needed}}
* Existence of weak solutions globally in time{{Citation needed}}
 
== References ==
 
{{reflist|refs=
<ref name="CMT">{{Citation | last1=Constantin | first1=Peter | last2=Majda | first2=Andrew J. | last3=Tabak | first3=Esteban | title=Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar | url=http://stacks.iop.org/0951-7715/7/1495 | year=1994 | journal=Nonlinearity | issn=0951-7715 | volume=7 | issue=6 | pages=1495–1533}}</ref>
<ref name="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
<ref name="ConstVicol">{{Citation | last1=Constantin | first1=Peter | last2=Vicol | first2=Vlad | title=Nonlinear maximum principles for dissipative linear nonlocal operators and applications | url=http://arxiv.org/abs/1110.0179 | year=2012 | journal=[[Geometric and Functional Analysis]],to appear}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Eventual regularization for the slightly supercritical quasi-geostrophic equation | url=http://dx.doi.org/10.1016/j.anihpc.2009.11.006 | doi=10.1016/j.anihpc.2009.11.006 | year=2010 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=27 | issue=2 | pages=693–704}}</ref>
<ref name="KNV">{{Citation | last1=Kiselev | first1=A. | last2=Nazarov | first2=F. | last3=Volberg | first3=A. | title=Global well-posedness for the critical 2D dissipative quasi-geostrophic equation | url=http://dx.doi.org/10.1007/s00222-006-0020-3 | doi=10.1007/s00222-006-0020-3 | year=2007 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=167 | issue=3 | pages=445–453}}</ref>
<ref name="KN">{{Citation | last1=Kiselev | first1=A. | last2=Nazarov | first2=F. | title=A variation on a theme of Caffarelli and Vasseur | year=2009 | journal=Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u\i Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI) | issn=0373-2703 | volume=370 | pages=58–72}}</ref>
<ref name="K">{{Citation | last1=Kiselev | first1=A. | title=Regularity and blow up for active scalars | url=http://dx.doi.org/10.1051/mmnp/20105410 | doi=10.1051/mmnp/20105410 | year=2010 | journal=Mathematical Modelling of Natural Phenomena | issn=0973-5348 | volume=5 | issue=4 | pages=225–255}}</ref>
<ref name="R">{{Citation | last1=Resnick | first1=Serge G. | title=Dynamical problems in non-linear advective partial differential equations | url=http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9542767 | publisher=ProQuest LLC, Ann Arbor, MI | year=1995}}</ref>
<ref name="D">{{Citation | last1=Dabkowski | first1=M. | title=Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2011 | journal=Geometric and Functional Analysis | issn=1016-443X | volume=21 | issue=1 | pages=1–13}}</ref>
<ref name="Y"> {{Citation | last1=Yu | first1=Xinwei | title=Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation | url=http://dx.doi.org/10.1016/j.jmaa.2007.06.064 | doi=10.1016/j.jmaa.2007.06.064 | year=2008 | journal=Journal of Mathematical Analysis and Applications | issn=0022-247X | volume=339 | issue=1 | pages=359–371}} </ref>
}}

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Using both footnote-style and bibliography-style references

== Content ==

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== References ==

{{Reflist}}

{{Refbegin}}
* reference 1
* reference 2
{{Refend}}

Font size

The font size should reduce to 90% for most browsers, but may appear to show at 100% for Internet Explorer and possibly other browsers.[1] As of December 21, 2010, the standard <references /> tag has the same font styling. The smaller font may be disabled through Special:Preferences → Gadgets → Disable smaller font sizes of elements such as Infoboxes, Navboxes and References lists.

Columns

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See also bug combining grouped references and columns

List styles

File:Imbox notice.png This is an experimental feature that may be subject to change.
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Multiple uses

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See also bug combining grouped references and columns

List-defined references

Further information: WP:LDR

As of September 2009, references may be defined within {{Reflist}} using |refs= and invoked within the content. There are new error messages associated with this update, documented at Help:Cite errors. As usual, groups can be used. Defined references must be used within the body; unused references will show an error message.

Example

This is reference 1.<ref name="refname1" group="groupname" />
This is reference 2.<ref name="refname2" group="groupname" />
This is reference 3.<ref name="refname3" group="groupname" />

{{Reflist|group="groupname"|refs=
<ref name="refname1" group="groupname">content1</ref>
<ref name="refname2" group="groupname">content2</ref>
<ref name="refname3" group="groupname">content3</ref>
}}

Result

This is reference 1.[groupname 1] This is reference 2.[groupname 2] This is reference 3.[groupname 3]

  1. ↑ content1
  2. ↑ content2
  3. ↑ content3

Technical details

Browser support for columns

CSS3 multiple column layout
browser support
Internet
Explorer		 Firefox Safari Chrome Opera
6 3 3.1 7 10.5
7 3.5 3.2 8 10.6
8 3.6 4 9 11
9 4 5 10 11.1
10

Multiple columns are generated by using CSS3, which is still in development; thus only browsers that properly support the multi-column property will show multiple columns with {{Reflist}}.[5][6]

These browsers support CSS3 columns:

These browsers do not support CSS3 columns:

  • Microsoft Trident-based browsers including Internet Explorer through to version 9[7]
  • Opera through to version 11

Supporting CSS

{{Reflist}} uses a CSS rule in MediaWiki:Common.css to set the font size:

  • div.reflist { font-size: 90%; }

One of the following classes is assigned by the template when either column count or column width is set:

  • .references-column-count and .references-column-count-n (where n is the numer of columns)
  • .references-column-width

The following CSS properties are utilized using the {{column-count}} and {{column-width}} templates:

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Customizing the view

Further information: Help:Reference display customization

By editing your CSS, the personal appearance of the reference list can be customized. From Preferences, select the Appearance tab, then on the selected skin select Custom CSS. After editing and saving, follow the instructions at the top of the page to purge. See Wikipedia:Skin#Customisation (advanced users) for more help.

Font size

The font size for all reference lists defaults to 90% of the standard size. To change it, add:

<source lang="css"> ol.references, div.reflist, div.refbegin { 

   font-size: 90%;

} </source>

Change 90% to the desired size.

Columns

To disable columns, add:

<source lang="css"> .references-column-count, .references-column-width { 

   column-count: 1 !important;
   column-width: auto !important;
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   -webkit-column-count: 1 !important;
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} </source>

Similarly, to force all reference lists to a number of columns, change the number.

Column dividers

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<source lang="css"> .references-column-count, .references-column-width { 

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} </source>

You can alter the appearance of the dividers by changing the values.

Bugs

Note that, whereas {{reflist|15em|group="e"}} works as expected, {{reflist|15em |group="e"}} (with an extra space) currently does not display in columns.

Perennial suggestions

Collapsing and scrolling

There have been a number of requests to add functionality for a collapsible or scrolling reference list. These requests have not been fulfilled to due to issues with readability, accessibility, and printing. The applicable guidelines are at MOS:SCROLL. Links between the inline cite and the reference list do not work when the reference list is enclosed in a collapsed box.

To display the reference list in a scrollbox or collapsed per user, see Help:Reference display customization.

For discussion on previous attempts to do this with a template, see the discussions for Scrollref and Refbox.

Including the section title

There have been suggestions to include section header markup such as ==References==. This is inadvisable because:

  • There is no standard section name; see WP:FNNR
  • When transcluded, the article will have an edit link that will confusingly open the template for editing

See also

References

  1. ↑ See User:Edokter/fonttest for a comparison of font sizes for various browsers; see previous discussions on changing the font size to resolve the IE issue.
  2. ↑ "Cascading Style Sheets Level 2 Revision 1 (CSS 2.1) Specification". W3C. December 7, 2010. http://www.w3.org/TR/2010/WD-CSS2-20101207/cover.html. 
  3. ↑ "Cascading Style Sheets, level 2 CSS2 Specification". W3C. April 11, 2008. http://www.w3.org/TR/2008/REC-CSS2-20080411/. 
  4. ↑ "CSS3 module: Lists". W3C. November 7, 2002. http://www.w3.org/TR/css3-lists/. 
  5. ↑ "CSS3 Multi-Column Thriller". December 30, 2005. http://www.stuffandnonsense.co.uk/archives/css3_multi-column_thriller.html. Retrieved November 24, 2006. 
  6. ↑ "CSS3 module: Multi-column layout". W3C. December 15, 2005. http://www.w3.org/TR/css3-multicol/. Retrieved November 24, 2006. 
  7. ↑ "CSS Compatibility and Internet Explorer: Multi-column Layout". Microsoft Developer Network. Microsoft. http://msdn.microsoft.com/en-us/library/cc351024(VS.85).aspx#multicolumn. Retrieved March 16, 2011. 

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