Surface quasi-geostrophic equation - Revision history

imported>Vlad: /* References */

2012-05-08T19:47:36Z

References

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	<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="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="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="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="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>

imported>Vlad: /* Critical case: $s=1/2$ */

2012-05-08T19:43:00Z

Critical case: $s=1/2$

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	=== Critical case: $s=1/2$ ===		=== Critical case: $s=1/2$ ===

	The equation is well posed globally. There are ~~three~~ known proofs.		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]].		* '''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]].		* '''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.		* '''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: $s<1/2$ ===		=== Supercritical case: $s<1/2$ ===

imported>Luis at 17:25, 25 April 2012

2012-04-25T17:25:51Z

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	Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$		Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$

	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]].		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.

	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.		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.

imported>Luis at 23:13, 5 February 2012

2012-02-05T23:13:32Z

← Older revision		Revision as of 18:13, 5 February 2012
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	The global well posedness of the equation is an open problem. Some partial results are known:		The global well posedness of the equation is an open problem. Some partial results are known:

	* Existence of solutions locally in time{{Citation needed}}		* Existence of smooth solutions locally in time{{Citation needed}}
			* Existence of weak solutions globally in time{{Citation needed}}

	== References ==		== References ==

imported>Luis: /* Sub-critical case: $s>1/2$ */

2011-06-01T20:18:53Z

Sub-critical case: $s>1/2$

← Older revision		Revision as of 15:18, 1 June 2011
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	=== Sub-critical case: $s>1/2$ ===		=== Sub-critical case: $s>1/2$ ===

	The equation is well posed globally. The proof can be done with ~~several~~ methods using only soft functional analysis or Fourier analysis.		The equation is well posed globally. The proof can be done with [[perturbation methods]] using only soft functional analysis or Fourier analysis.

	=== Critical case: $s=1/2$ ===		=== Critical case: $s=1/2$ ===

imported>Luis: /* Scaling and criticality */

2011-06-01T20:18:06Z

Scaling and criticality

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	If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.		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 ~~equation~~. For ~~smaller~~ values of $s$, the diffusion ~~is stronger than~~ the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.		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 ==		== Well posedness results ==

imported>Luis at 05:30, 31 May 2011

2011-05-31T05:30:24Z

← Older revision		Revision as of 00:30, 31 May 2011
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	The global well posedness of the equation is an open problem. Some partial results are known:		The global well posedness of the equation is an open problem. Some partial results are known:

	* Existence of solutions locally in time~~???~~		* Existence of solutions locally in time{{Citation needed}}

	== References ==		== References ==

69.217.124.45 at 04:29, 31 May 2011

2011-05-31T04:29:46Z

← Older revision		Revision as of 23:29, 30 May 2011
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	* '''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]].		* '''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]].		* '''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 for general [[drift-diffusion equations]] as well.		* '''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.

	=== Supercritical case: $s<1/2$ ===		=== Supercritical case: $s<1/2$ ===
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	The global well posedness of the equation is an open problem. Some partial results are known:		The global well posedness of the equation is an open problem. Some partial results are known:

	* ???		* Existence of solutions locally in time???

	== References ==		== References ==

69.217.124.45 at 04:28, 31 May 2011

2011-05-31T04:28:16Z

← Older revision		Revision as of 23:28, 30 May 2011
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	* '''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]].		* '''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 $\theta$.		* '''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.		* '''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 for general [[drift-diffusion equations]] as well.

	=== Supercritical case: $s<1/2$ ===		=== Supercritical case: $s<1/2$ ===

imported>Luis: moved Surface quasi-geostrophic equations to Surface quasi-geostrophic equation

2011-05-27T04:07:48Z

moved Surface quasi-geostrophic equations to Surface quasi-geostrophic equation

← Older revision	Revision as of 23:07, 26 May 2011
(No difference)

← Older revision		Revision as of 14:47, 8 May 2012
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	<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="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="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="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="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>