nonlocal pde - User contributions [en]

To Do List

2011-05-31T05:19:54Z

69.217.124.45:

== Current Projects / To do list ==

We need to come up with some organization for the articles.

The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.

* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron method]] and uniqueness through the [[comparison principle]].

* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].

* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].

* A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]].

* Some discussion on [[models]] involving [[Levy processes]] and [[stochastic control]].

* Some references to equations from fluids including the [[surface quasi-geostrophic equation]].

* A page about [[semilinear equations]] including the [[surface quasi-geostrophic equation]] and also some form of KPP.

* [[drift-diffusion equations]].

* [[Nonlocal minimal surfaces]].

* Fractional curvatures in conformal geometry.

We may want to include a [[mini second order elliptic wiki]] inside this wiki.

We may want to have a section on [[open problems]], for example integral ABP for general kernels, harnack for kernels with a
less restrictive bound below, C^{1,alpha} estimate in bounded domains with a nonsmooth kernel, supercritical quasi-geostrophic,
classification of nonlocal minimal cones, etc...

We may want to have a list of [[upcoming events]] such as conferences, workshops, summer schools, etc.

Drift-diffusion equations

2011-05-31T05:13:21Z

69.217.124.45:

A drift-(fractional)diffusion equation refers to an evolution equation of the form
\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
where $b$ is any vector fields. The stationary version can also be of interest
\[ b \cdot \nabla u + (-\Delta)^s u = 0.\]

This type of equations under several contexts. It is often useful to apply regularity results about drift-diffusion equations to semilinear equations from fluid dynamics (for example the [[surface quasi-geostrophic equations]]). For this reason, the assumption that $\mathrm{div} \ b = 0$ is taken sometimes. The equation can also be derived as the flow of a vector field with $\alpha$-stable white noise.

There are a number of regularity results depending of the assumptions on the vector field $b$ and the values of the exponent $s$. The type of regularity results available for drift-diffusion equations can be separated into two categories depending on whether or not they are proved using [[perturbation methods]]. Perturbative results are characterized by an assumption on the vector field $b$ that implies that at small scales, its flow is negligible in comparison with fractional diffusion.

== Scaling ==

The terms supercritical, critical, and subcritical are often used to denote whether the diffusion part of the equation controls the regularity or not. Given a quantitative assumption on the vector field $b$, one can check if it is subcritical, critical, or supercritical by checking the effect of scaling. More precisely, we know that the rescaled function $u_\lambda(t,x) = u(\lambda^{2s}t,\lambda x)$ satisfies the equation
\[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]

If an a priori estimate on $b$ improves with the scaling $\lambda^{2s-1} b(\lambda^{2s}t,\lambda x)$ for $\lambda <1$, the equation is in the '''subcritical''' regime. Typically, a regularity result for a drift-diffusion equation with subcritical assumptions on $b$ would be obtained using [[perturbation methods]]. Sometimes a strong regularity result holds and the solutions are necessarily classical.

If an a priori estimate on $b$ is invariant by the scaling $\lambda^{2s-1} b(\lambda^{2s}t,\lambda x)$, the equation is in the '''critical''' regime. There are some regularity results for drift-diffusion equation with critical assumptions on $b$ but the proofs are more delicate and cannot be obtained via perturbation methods.

If an a priori estimate on $b$ deteriorates with the scaling $\lambda^{2s-1} b(\lambda^{2s}t,\lambda x)$ for $\lambda <1$, the equation is in the '''supercritical''' regime. There is no regularity result available for any kind of supercritical assumption on $b$. In this case, the transport part of the equation is expected to dominate the equation.

== Pertubative results ==

=== Kato classes ===
The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller.

=== $C^{1,\alpha}$ estimates ===
Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.

== Scale invariant results ==

=== Divergence-free vector fields ===
If the vector field $b$ is divergence free, the method of [[De Giorgi-Nash-Moser]] can be adapted to show that the solution $u$ becomes immediately Holder continuous.

In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(BMO)$ <ref name="CV"/> <ref name="KN"/>.

In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ <ref name="CW"/>.

=== Vector fields with arbitrary divergence ===
For any bounded vector field $b$, one method for obtaining [[Holder estimates]] for integro-differential equations can be used to show that $u$ becomes immediately Holder continuous.

In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(L^\infty)$ <ref name="S1"/>, which is a marginally stronger assumption than the one needed if $b$ was assumed divergence-free.

In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ <ref name="S2"/>, which is the same assumption than the one needed if $b$ was assumed divergence-free. Therefore, if $s<1/2$, there is no known advantage in assuming $\mathrm{div} \ b =0$.

== References ==
{{reflist|refs=
<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="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="CW">{{Citation | last1=Constantin | first1=Peter | last2=Wu | first2=Jiahong | title=Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations | url=http://dx.doi.org/10.1016/j.anihpc.2007.10.002 | doi=10.1016/j.anihpc.2007.10.002 | year=2009 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=26 | issue=1 | pages=159–180}}</ref>
<ref name="S1">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion | url=http://dx.doi.org/10.1016/j.aim.2010.09.007 | doi=10.1016/j.aim.2010.09.007 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=2020–2039}}</ref>
<ref name="S2">{{Citation | last1=Silvestre | first1=Luis | title=Holder estimates for advection fractional-diffusion equations | year=To appear | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze}}</ref>
}}

2011-05-28T22:35:16Z

69.217.124.45:

* For general [[fully nonlinear integro-differential equations]], interior [[differentiability estimates|$C^{1,\alpha}$ estimates]] can be proved in a variety of situations. The simplest assumption would be for a translation invariant [[fully nonlinear integro-differential equations|uniformly elliptic]] equations with respect to the [[Linear integro-differential operator|class of kernels]] that are uniformly elliptic of order $s$ and in the smoothness class of order 1 <ref name="CS"/>. There are several other [[differentiability estimates|$C^{1,\alpha}$ estimates]] for variations of this situation (smooth coefficients, kernels close to the smoothness class, etc...) <ref name="CS2"/>.

* A [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]] says that for the [[Bellman equation]], for a family of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ <ref name="CS3"/>. This is enough regularity for the solutions to be [[classical solutions|classical]].

== References ==
{{reflist|refs=
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="CS2">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity results for nonlocal equations by approximation | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–30}}</ref>
<ref name="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</ref>
}}

Regularity results for fully nonlinear integro-differential equations

2011-05-28T22:23:23Z

69.217.124.45: Created page with "* For general fully nonlinear integro-differential equations, interior $C^{1,\alpha}$ estimates can be proved in a variety of situations. The ..."

Surface quasi-geostrophic equation

2011-05-24T01:43:13Z

69.217.124.45:

$
\newcommand{\R}{\mathbb{R}}
$

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.

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 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.

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.

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" />.

== Conserved quantities ==

The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).

* ''' 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 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.

== Well posedness results ==

=== 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.

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

The equation is well posed globally. There are three 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 $\theta$.
* '''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.

=== 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:

* ???

== 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="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>
}}

Surface quasi-geostrophic equation

2011-05-24T01:39:23Z

69.217.124.45:

$
\newcommand{\R}{\mathbb{R}}
$

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.

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 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.

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.

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" />.

== Conserved quantities ==

The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).

* ''' 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 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.

== Well posedness results ==

=== 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.

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

The equation is well posed globally. There are three 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 $\theta$.
* '''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.

=== 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.
* 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:

* ???

== 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="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>
}}

Surface quasi-geostrophic equation

2011-05-24T01:38:14Z

69.217.124.45:

$
\newcommand{\R}{\mathbb{R}}
$

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.

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 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.

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.

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">.

== Conserved quantities ==

The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).

* ''' 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 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.

== Well posedness results ==

=== 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.

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

The equation is well posed globally. There are three 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 $\theta$.
* '''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.

=== 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.
* 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:

* ???

== 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="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>
}}

Hölder estimates

2011-05-24T01:14:38Z

69.217.124.45:

Holder continuity of the solutions can sometimes be proved only from ellipticity assumptions on the equation, without depending on smoothness of the coefficients. This allows great flexibility in terms of applications of the result. The corresponding result for elliptic equations of second order is the [[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser]] theorem in the divergence form.

The holder estimates are closely related to the [[Harnack inequality]].

There are integro-differential versions of both [[De Giorgi-Nash-Moser]] theorem and [[Krylov-Safonov]] theorem. The former uses variational techniques and is stated in terms of Dirichlet forms. The latter is based on comparison principles.

A Holder estimate says that a solution to an [[integro-differential equation]] $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small). It is '''very important''' to allow for a very rough dependence of $L_x$ with respect to $x$, since the result then applies to the linearization of fully nonlinear equations without any extra a priori estimate. On the other hand, the linearization of a fully nonlinear equation (for example the [[Isaacs equation]]) would inherit the initial assumptions regarding for the kernels with respect to $y$. Therefore, smoothness (or even structural) assumptions for the kernels with respect to $y$ can be made keeping such result useful.

In the non variational setting the integro-differential operators $L_x$ will be assumed to belong to some family, but no continuity is assumed for its dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$
Since [[integro-differential equations]] allow for a great flexibility of equations, there are several variations on the result: different assumptions on the kernels, mixed local terms, evolution equations, etc. The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for the [[extremal operators]] corresponding to the family of operators $L$, which stresses the nonlinear character of the estimates.

Some estimates blow up as the order of the equation converges to two, and others pass to the limit. In the latter case the estimates are a true generalization of either [[De Giorgi-Nash-Moser]] or [[Krylov-Safonov]]]

== Estimates which blow up as the order goes to two ==

=== Non variational case ===

The Holder estimates were first obtained using probabilistic techniques <ref name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to a family satisfying certain set of assumptions. No regularity of any kind is assumed for $K$ with respect to $x$. The assumption for the family of operators are
# '''Scaling''': If $L$ belongs to the family, then so does its scaled version $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
# '''Nondegeneracy''': If $K$ is the kernel associated to $L$, $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\sup_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.

The right hand side $f$ is assumed to belong to $L^\infty$.

A particular cases in which this result applies is the uniformly elliptic case.
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no continuity of $s$ respect to $x$ is required.
The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$. However this assumption can be overcome in the following two situations.
* For $s<1$, the symmetry assumption can be removed if the equation does not contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy = f(x)$ in $B_1$.
* For $s>1$, the symmetry assumption can be removed if the drift correction term is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy = f(x)$ in $B_1$.

The reason for the symmetry assumption, or the modification of the drift correction term, is that in the original formulation the term $y \cdot \nabla u(x) \, \chi_{B_1}(y)$ is not scale invariant.

=== Variational case ===

A [[Dirichlet forms]] is a quadratic functional of the form
$$ J(u) = \iint_{\R^n \times \R^n} |u(x)-u(y)|^2 K(x,y) \, dx \, dy $$.

Minimizers of Dirichlet forms are a nonlocal version of minimizers of integral functionals as in [[De Giorgi-Nash-Moser]] theorem.

The symmetry assumption $K(x,y)=K(y,x)$ is natural since the skew-symmetric part of $K$ would be ignored by the quadratic functional.

It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous <ref name="CCV"/>. The method of the proof builds an integro-differential version of the parabolic De Giorgi technique that was developed for the study of critical [[surface quasi-geostrophic equation]].

At some point in the original proof of De Giorgi, it is used that the characteristic functions of a set of positive measure do not belong to $H^1$. Moreover, a quantitative estimate is required about the measure of ''intermediate'' level sets for $H^1$ functions. In the integro-differential context, the required statement to carry out the proof would be the same with the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would even require a non trivial proof for $s$ close to $2$. The difficulty is bypassed though an argument that takes advantage of the nonlocal character of the equation, and hence the estimate blows up as the order approaches two.

== Estimates which pass to the second order limit ==

=== Non variational case ===

An integro-differential generalization of [[Krylov-Safonov]] theorem is available <ref name="CS"/>. The assumption on the kernels are
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.

The right hand side $f$ is assumed to be in $L^\infty$. The constants in the Holder estimate do not blow up as $s \to 2$.

=== Variational case ===

In the stationary case, it is known that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser]] to the nonlocal setting <ref name="K"/>.

== References ==
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Hölder continuity of harmonic functions with respect to operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 | doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=30 | issue=7 | pages=1249–1259}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</ref>
}}

Hölder estimates

2011-05-24T01:12:31Z

69.217.124.45:

Holder continuity of the solutions can sometimes be proved only from ellipticity assumptions on the equation, without depending on smoothness of the coefficients. This allows great flexibility in terms of applications of the result. The corresponding result for elliptic equations of second order is the [[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser]] theorem in the divergence form.

The holder estimates are closely related to the [[Harnack inequality]].

There are integro-differential versions of both [[De Giorgi-Nash-Moser]] theorem and [[Krylov-Safonov]] theorem. The former uses variational techniques and is stated in terms of Dirichlet forms. The latter is based on comparison principles.

A Holder estimate says that a solution to an [[integro-differential equation]] $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small). It is '''very important''' to allow for a very rough dependence of $L_x$ with respect to $x$, since the result then applies to the linearization of fully nonlinear equations without any extra a priori estimate. On the other hand, the linearization of a fully nonlinear equation (for example the [[Isaacs equation]]) would inherit the initial assumptions regarding for the kernels with respect to $y$. Therefore, smoothness (or even structural) assumptions for the kernels with respect to $y$ can be made keeping such result useful.

In the non variational setting the integro-differential operators $L_x$ will be assumed to belong to some family, but no continuity is assumed for its dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$
Since [[integro-differential equations]] allow for a great flexibility of equations, there are several variations on the result: different assumptions on the kernels, mixed local terms, evolution equations, etc. The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for the [[extremal operators]] corresponding to the family of operators $L$, which stresses the nonlinear character of the estimates.

Some estimates blow up as the order of the equation converges to two, and others pass to the limit. In the latter case the estimates are a true generalization of either [[De Giorgi-Nash-Moser]] or [[Krylov-Safonov]]]

== Estimates which blow up as the order goes to two ==

=== Non variational case ===

The Holder estimates were first obtained using probabilistic techniques <ref name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to a family satisfying certain set of assumptions. No regularity of any kind is assumed for $K$ with respect to $x$. The assumption for the family of operators are
# '''Scaling''': If $L$ belongs to the family, then so does its scaled version $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
# '''Nondegeneracy''': If $K$ is the kernel associated to $L$, $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\sup_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.

The right hand side $f$ is assumed to belong to $L^\infty$.

A particular cases in which this result applies is the uniformly elliptic case.
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no continuity of $s$ respect to $x$ is required.
The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$. However this assumption can be overcome in the following two situations.
* For $s<1$, the symmetry assumption can be removed if the equation does not contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy = f(x)$ in $B_1$.
* For $s>1$, the symmetry assumption can be removed if the drift correction term is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy = f(x)$ in $B_1$.

The reason for the symmetry assumption, or the modification of the drift correction term, is that in the original formulation the term $y \cdot \nabla u(x) \, \chi_{B_1}(y)$ is not scale invariant.

=== Variational case ===

A [[Dirichlet forms]] is a quadratic functional of the form
$$ J(u) = \iint_{\R^n \times \R^n} |u(x)-u(y)|^2 K(x,y) \, dx \, dy $$.

Minimizers of Dirichlet forms are a nonlocal version of minimizers of integral functionals as in [[De Giorgi-Nash-Moser]] theorem.

The symmetry assumption $K(x,y)=K(y,x)$ is natural since the skew-symmetric part of $K$ would be ignored by the quadratic functional.

It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous <ref name="CCV"/>. The method of the proof builds an integro-differential version of the parabolic De Giorgi technique that was developed for the study of critical [[surface quasi-geostrophic equation]].

At some point in the original proof of De Giorgi, it is used that the characteristic functions of a set of positive measure do not belong to $H^1$. Moreover, a quantitative estimate is required about the measure of ''intermediate'' level sets for $H^1$ functions. In the integro-differential context, the required statement to carry out the proof would be the same with the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would even require a non trivial proof for $s$ close to $2$. The difficulty is bypassed though an argument that takes advantage of the nonlocal character of the equation, similarly as in {{Citation needed}}.

== Estimates which pass to the second order limit ==

=== Non variational case ===

An integro-differential generalization of [[Krylov-Safonov]] theorem is available <ref name="CS"/>. The assumption on the kernels are
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.

The right hand side $f$ is assumed to be in $L^\infty$. The constants in the Holder estimate do not blow up as $s \to 2$.

=== Variational case ===

In the stationary case, it is known that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser]] to the nonlocal setting <ref name="K"/>.

== References ==
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Hölder continuity of harmonic functions with respect to operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 | doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=30 | issue=7 | pages=1249–1259}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</ref>
}}