Surface quasi-geostrophic equation and Open problems: Difference between pages

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== Well posedness of the supercritical [[surface quasi-geostrophic equation]] ==
  \newcommand{\R}{\mathbb{R}}
Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s \in (0,1/2)$. Is there a global classical solution $\theta :\R^2 \to \R$ for the SQG equation?
$
\begin{align*}
\theta(x,0) &= \theta_0(x) \\
\theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
\end{align*}
where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.


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.
This is a very difficult open problem. It is believed that a solution would be a major step towards the understanding of Navier-Stokes equation. In the supercritical regime $s\in (0,1/2)$, the effect if the drift term is larger than the diffusion in small scales. Therefore, it seems unlikely that a proof of well posedness could be achieved with the methods currently known and listed in this wiki.


Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$
Note that if the relation between $u$ and $\theta$ was changed by $u = R\theta$, then the equation is ill posed. This suggests that the divergence free nature of $u$ must play an important role, unlike the critical and subcritical cases $s \geq 1/2$.


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.
== Regularity of [[nonlocal minimal surfaces]] ==


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.
A nonlocal minimal surface that is sufficiently flat is known to be smooth <ref name="CRS"/>. The possibility of singularities in the general case reduces to the analysis of a possible existence of nonlocal minimal cones. The problem can be stated as follows.


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.
For any $s \in (0,1)$, and any natural number $n$, is there any set $A \in \R^n$, other than a half space, such that
# $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
# If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]


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" />.
When $s$ is sufficiently close to one, such set does not exist if $n < 8$.


== An integral ABP estimate ==


== Conserved quantities ==
The nonlocal version of the [[Alexadroff-Bakelman-Pucci estimate]] holds either for a right hand side in $L^\infty$ <ref name="CS"/> (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels <ref name="GS"/>. Would the following result be true?


The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).
Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
\[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
Where $\chi_{A_n}$ stands for the characteristic function of the sets $A_n$. Assume that the kernels $K$ satisfy symmetry and a uniform ellipticity condition
\begin{align*}
K(x,y) &= K(x,-y) \\
\lambda |y|^{-n-s} \leq K(x,y) &\leq \Lambda |y|^{-n-s} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
\end{align*}
If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?


* ''' Maximum principle '''
This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/>


The supremum of $\theta$ occurs at time zero: $||\theta(t,.)||_{L^\infty} \leq ||\theta(0,.)||_{L^\infty}$.
== A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form ==
Consider two continuous functions $u$ and $v$ such that
\begin{align*}
u(x) &\leq v(x) \qquad \text{for all $x$ outside some set } \Omega,\\
F(x,\{I_\alpha u(x)\}) &\geq F(x,\{I_\alpha v(x)\})\qquad \text{for all $x \in \Omega$}.
\end{align*}
Is it true that $u \leq v$ in $\Omega$ as well?


* '''Conservation of energy'''.
It is natural to expect this result to hold if $F$ is continuous respect to $x$ and the [[linear integro-differential operators]] $I_\alpha$ satisfy some nondegeneracy condition and continuity respect to $x$, e.g.
\begin{align*}
I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz)
\end{align*}
where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with respect to $x$?
Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.<ref name="BI"/>


A classical solution $u$ satisfies the energy equality
== A local [[differentiability estimates|$C^{1,\alpha}$ estimate]] for integro-differential equations with nonsmooth kernels ==
$$ \int_{\R^2} \theta(0,x)^2 \ dx = \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$


In the case of weak solutions, only the energy inequality is available
Assume that $u : \R^n \to \R$ is a bounded function satisfying a [[fully nonlinear integro-differential equation]] $Iu=0$ in $B_1$. Assume that $I$ is elliptic with respect to the family of kernels $K$ such that
$$ \int_{\R^2} \theta(0,x)^2 \ dx \geq \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$
\[ \frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}. \]
Is it true that $u \in C^{1,\alpha}(B_1)$?


* '''$H^{-1/2}$ estimate'''
An extra symmetry assumptions on the kernels may or maynot be necessary. The difficulty here is the lack of any smoothness assumption on the tails of the kernels $K$. This assumption is used in a localization argument in the proof of the [[differentiability estimates|$C^{1,\alpha}$ estimates]] <ref name="CS"/>. It is conceivable that the assumption may not be necessary at least for $s>1$.


The $H^{-1/2}$ norm of $\theta$ does not increase in time.
The need of the smoothness assumption for the $C^{1,\alpha}$ estimate is a subtle technical requirement. It is easy to overlook going through the proof naively.


$$ \int_{\R^2} |(-\Delta)^{-1/4} \theta(0,x)|^2 \ dx = \int_{\R^2} |(-\Delta)^{-1/4}\theta(t,x)|^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2-1/4}\theta(r,x)|^2 \ dx \ dr.$$
Note that the assumption is used only to localize an iteration of the [[Holder estimates]]. An equation of the form $Iu = f$ in the whole space $\R^n$ with $f$ smooth enough would easily have $C^{1,\alpha}$ estimates without any smoothness restriction of the tails of the kernel.


It is not clear how important or difficult this problem is. The solution may end up being a relatively simple technical approximation technique or may require a fundamentally new idea.


== Scaling and criticality ==
The same difficulty arises for $C^{s+\alpha}$ [[nonlocal Evans-Krylov theorem|estimates for convex equations]]. For example, is it true that a bounded function $u$ such that $M^+u = 0$ in $B_1$, where $M^+$ is the [[extremal operators|monster Pucci operator]] is $C^{s+\alpha}$ for some $\alpha>0$?


If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.
== A nonlocal generalization of the parabolic [[Krylov-Safonov theorem]] ==


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.
Let $u$ be a bounded function in $\R^n \times [-1,0]$ such that it solves an integro-differential parabolic equation
\[ u_t - \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y = 0 \qquad \text{in } B_1 \times (-1,0).\]
Making the usual symmetry and uniform ellipticity assumptions on the kernel $K$:
\begin{align*}
K(x,y) &= K(x,-y) \\
\frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(x,y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
\end{align*}
Is it true that the solutions $u$ is Holder continuous in $B_{1/2} \times [-1/2,0]$, with an estimate
\[ ||u||_{C^\alpha(B_{1/2} \times [-1/2,0])} \leq C ||u||_{L^\infty(\R^n \times [-1,0])}, \]
for constants $C$ and $\alpha>0$ which do not blow up as $s \to 2$?


== Well posedness results ==
For an estimate with constants that blow up as $s \to 2$, one can easily adapt an argument for [[drift-diffusion equations]] <ref name="S2"/>.


=== Sub-critical case: $s>1/2$ ===
The elliptic version of this result is well known <ref name="CS"/>. The proof is not easy to adapt to the parabolic case because the [[Alexadroff-Bakelman-Pucci estimate]] is quite different in the elliptic and parabolic case.


The equation is well posed globally. The proof can be done with [[perturbation methods]] using only soft functional analysis or Fourier analysis.
For gradient flows of Dirichlet forms, the problems appears open as well. However, it is conceivable that one could adapt the proof of the stationary case <ref name="K"/> to obtain the result without a major difficulty.  


=== Critical case: $s=1/2$ ===
== Optimal regularity for the [[obstacle problem]] for a general integro-differential operator ==


The equation is well posed globally. There are four known proofs.
Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
\begin{align*}
u &\geq \varphi \qquad \text{in } \R^n, \\
(-\Delta)^{s/2} u &\geq 0 \qquad \text{in } \R^n, \\
(-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\
\end{align*}
where $\varphi$ is a smooth compactly supported function. It is known that $u \in C^{1,s/2}$ (where $s$ coincides with the order of the fractional Laplacian). This regularity is optimal in the sense that one can construct solutions that are not in $C^{1,s/2+\varepsilon}$ for any $\varepsilon>0$. One can consider the same problem replacing the fractional Laplacian by any other nonlocal operator. In fact, this problem corresponds to the [[optimal stopping problem]] in stochastic control, with applications to mathematical finance. The fractional Laplacian is just the particular case when the [[Levy  process]] involved is $\alpha$-stable. The optimal regularity for the general problem is currently an open problem. Even in the linear case with constant coefficients this is nontrivial. If $u$ is a solution of
\begin{align*}
u &\geq \varphi \qquad \text{in } \R^n, \\
L u &\leq 0 \qquad \text{in } \R^n, \\
L u &= 0 \qquad \text{in } \{u>\varphi\}, \\
\end{align*}
where $L$ is a [[linear integro-differential operator]], then what is the optimal regularity we can obtain for $u$?


* '''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]].
The optimal regularity would naturally depend on some assumptions on the linear operator $L$. If $L$ is a purely integro-differential with a kernel $K$ satisfying the usual ellipticity conditions
* '''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]].
\begin{align*}
* '''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.
K(y) &= K(-y) \\
* '''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.
\frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2),
\end{align*}
it is natural to expect the solution $u$ to be $C^s$, but this regularity is not optimal. Is the optimal regularity going to be $C^{1,s/2}$ as in the fractional Laplacian case? Most probably some extra assumption on the kernel will be needed.


=== Supercritical case: $s<1/2$ ===
A solution to this problem would be very interesting if it provides an optimal regularity result for a natural family of kernels. If the assumption is something hard to check (like for example that there exists an extension problem whose Dirichlet to Neumann map is $L$), then the result may not be that interesting.


The global well posedness of the equation is an open problem. Some partial results are known:
== Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==


* Existence of solutions locally in time.
Consider a [[drift-diffusion equation]] of the form
* Existence of global weak solutions. <ref name="R"/>
\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]
* 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 solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty(BMO)$ <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).


The global well posedness of the equation is an open problem. Some partial results are known:
The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.


* Existence of smooth solutions locally in time{{Citation needed}}
== Complete classification of free boundary points in the [[fractional obstacle problem]] ==
* Existence of weak solutions globally in time{{Citation needed}}
 
Some free boundary points of the [[fractional obstacle problem]] are classified as regular and the free boundary is known to be smooth around them <ref name="CSS"/>. Other points on the free boundary are classified as singular, and they are shown to be contained in a lower dimensional differentiable surface, and therefore to be rare <ref name="GP"/>. However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\
# Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the [[thin obstacle problem]], using the [[extension technique]]. However, it is not clear if such examples can be made in the original formulation of the [[fractional obstacle problem]] since because of the decay at infinity requirement.
# In case that point of a third category exist, is the free boundary smooth around these points in the ''third category''?
 
Other open problems concerning the [[fractional obstacle problem]] are
# Further regularity of the free boundary in smoother classes than $C^{1,\alpha}$.
# Regularity of the free boundary for the parabolic problem.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
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}}
}}

Revision as of 21:44, 23 February 2012

Well posedness of the supercritical surface quasi-geostrophic equation

Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s \in (0,1/2)$. Is there a global classical solution $\theta :\R^2 \to \R$ for the SQG equation? \begin{align*} \theta(x,0) &= \theta_0(x) \\ \theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty) \end{align*} where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.

This is a very difficult open problem. It is believed that a solution would be a major step towards the understanding of Navier-Stokes equation. In the supercritical regime $s\in (0,1/2)$, the effect if the drift term is larger than the diffusion in small scales. Therefore, it seems unlikely that a proof of well posedness could be achieved with the methods currently known and listed in this wiki.

Note that if the relation between $u$ and $\theta$ was changed by $u = R\theta$, then the equation is ill posed. This suggests that the divergence free nature of $u$ must play an important role, unlike the critical and subcritical cases $s \geq 1/2$.

Regularity of nonlocal minimal surfaces

A nonlocal minimal surface that is sufficiently flat is known to be smooth [1]. The possibility of singularities in the general case reduces to the analysis of a possible existence of nonlocal minimal cones. The problem can be stated as follows.

For any $s \in (0,1)$, and any natural number $n$, is there any set $A \in \R^n$, other than a half space, such that

  1. $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
  2. If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds

\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]

When $s$ is sufficiently close to one, such set does not exist if $n < 8$.

An integral ABP estimate

The nonlocal version of the Alexadroff-Bakelman-Pucci estimate holds either for a right hand side in $L^\infty$ [2] (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels [3]. Would the following result be true?

Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$, \[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \] Where $\chi_{A_n}$ stands for the characteristic function of the sets $A_n$. Assume that the kernels $K$ satisfy symmetry and a uniform ellipticity condition \begin{align*} K(x,y) &= K(x,-y) \\ \lambda |y|^{-n-s} \leq K(x,y) &\leq \Lambda |y|^{-n-s} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2). \end{align*} If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?

This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.[3]

A comparison principle for $x$-dependent nonlocal equations which are not in the Levy-Ito form

Consider two continuous functions $u$ and $v$ such that \begin{align*} u(x) &\leq v(x) \qquad \text{for all $x$ outside some set } \Omega,\\ F(x,\{I_\alpha u(x)\}) &\geq F(x,\{I_\alpha v(x)\})\qquad \text{for all $x \in \Omega$}. \end{align*} Is it true that $u \leq v$ in $\Omega$ as well?

It is natural to expect this result to hold if $F$ is continuous respect to $x$ and the linear integro-differential operators $I_\alpha$ satisfy some nondegeneracy condition and continuity respect to $x$, e.g. \begin{align*} I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz) \end{align*} where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with respect to $x$?

Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.[4]

A local $C^{1,\alpha}$ estimate for integro-differential equations with nonsmooth kernels

Assume that $u : \R^n \to \R$ is a bounded function satisfying a fully nonlinear integro-differential equation $Iu=0$ in $B_1$. Assume that $I$ is elliptic with respect to the family of kernels $K$ such that \[ \frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}. \] Is it true that $u \in C^{1,\alpha}(B_1)$?

An extra symmetry assumptions on the kernels may or maynot be necessary. The difficulty here is the lack of any smoothness assumption on the tails of the kernels $K$. This assumption is used in a localization argument in the proof of the $C^{1,\alpha}$ estimates [2]. It is conceivable that the assumption may not be necessary at least for $s>1$.

The need of the smoothness assumption for the $C^{1,\alpha}$ estimate is a subtle technical requirement. It is easy to overlook going through the proof naively.

Note that the assumption is used only to localize an iteration of the Holder estimates. An equation of the form $Iu = f$ in the whole space $\R^n$ with $f$ smooth enough would easily have $C^{1,\alpha}$ estimates without any smoothness restriction of the tails of the kernel.

It is not clear how important or difficult this problem is. The solution may end up being a relatively simple technical approximation technique or may require a fundamentally new idea.

The same difficulty arises for $C^{s+\alpha}$ estimates for convex equations. For example, is it true that a bounded function $u$ such that $M^+u = 0$ in $B_1$, where $M^+$ is the monster Pucci operator is $C^{s+\alpha}$ for some $\alpha>0$?

A nonlocal generalization of the parabolic Krylov-Safonov theorem

Let $u$ be a bounded function in $\R^n \times [-1,0]$ such that it solves an integro-differential parabolic equation \[ u_t - \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y = 0 \qquad \text{in } B_1 \times (-1,0).\] Making the usual symmetry and uniform ellipticity assumptions on the kernel $K$: \begin{align*} K(x,y) &= K(x,-y) \\ \frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(x,y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2). \end{align*} Is it true that the solutions $u$ is Holder continuous in $B_{1/2} \times [-1/2,0]$, with an estimate \[ ||u||_{C^\alpha(B_{1/2} \times [-1/2,0])} \leq C ||u||_{L^\infty(\R^n \times [-1,0])}, \] for constants $C$ and $\alpha>0$ which do not blow up as $s \to 2$?

For an estimate with constants that blow up as $s \to 2$, one can easily adapt an argument for drift-diffusion equations [5].

The elliptic version of this result is well known [2]. The proof is not easy to adapt to the parabolic case because the Alexadroff-Bakelman-Pucci estimate is quite different in the elliptic and parabolic case.

For gradient flows of Dirichlet forms, the problems appears open as well. However, it is conceivable that one could adapt the proof of the stationary case [6] to obtain the result without a major difficulty.

Optimal regularity for the obstacle problem for a general integro-differential operator

Let $u$ be the solution to the obstacle problem for the fractional laplacian, \begin{align*} u &\geq \varphi \qquad \text{in } \R^n, \\ (-\Delta)^{s/2} u &\geq 0 \qquad \text{in } \R^n, \\ (-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\ \end{align*} where $\varphi$ is a smooth compactly supported function. It is known that $u \in C^{1,s/2}$ (where $s$ coincides with the order of the fractional Laplacian). This regularity is optimal in the sense that one can construct solutions that are not in $C^{1,s/2+\varepsilon}$ for any $\varepsilon>0$. One can consider the same problem replacing the fractional Laplacian by any other nonlocal operator. In fact, this problem corresponds to the optimal stopping problem in stochastic control, with applications to mathematical finance. The fractional Laplacian is just the particular case when the Levy process involved is $\alpha$-stable. The optimal regularity for the general problem is currently an open problem. Even in the linear case with constant coefficients this is nontrivial. If $u$ is a solution of \begin{align*} u &\geq \varphi \qquad \text{in } \R^n, \\ L u &\leq 0 \qquad \text{in } \R^n, \\ L u &= 0 \qquad \text{in } \{u>\varphi\}, \\ \end{align*} where $L$ is a linear integro-differential operator, then what is the optimal regularity we can obtain for $u$?

The optimal regularity would naturally depend on some assumptions on the linear operator $L$. If $L$ is a purely integro-differential with a kernel $K$ satisfying the usual ellipticity conditions \begin{align*} K(y) &= K(-y) \\ \frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2), \end{align*} it is natural to expect the solution $u$ to be $C^s$, but this regularity is not optimal. Is the optimal regularity going to be $C^{1,s/2}$ as in the fractional Laplacian case? Most probably some extra assumption on the kernel will be needed.

A solution to this problem would be very interesting if it provides an optimal regularity result for a natural family of kernels. If the assumption is something hard to check (like for example that there exists an extension problem whose Dirichlet to Neumann map is $L$), then the result may not be that interesting.

Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$)

Consider a drift-diffusion equation of the form \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]

The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty(BMO)$ [7]. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) [8] [9]. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).

The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur [7] can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.

Complete classification of free boundary points in the fractional obstacle problem

Some free boundary points of the fractional obstacle problem are classified as regular and the free boundary is known to be smooth around them [10]. Other points on the free boundary are classified as singular, and they are shown to be contained in a lower dimensional differentiable surface, and therefore to be rare [11]. However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\

  1. Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the thin obstacle problem, using the extension technique. However, it is not clear if such examples can be made in the original formulation of the fractional obstacle problem since because of the decay at infinity requirement.
  2. In case that point of a third category exist, is the free boundary smooth around these points in the third category?

Other open problems concerning the fractional obstacle problem are

  1. Further regularity of the free boundary in smoother classes than $C^{1,\alpha}$.
  2. Regularity of the free boundary for the parabolic problem.

References

  1. Caffarelli, Luis A.; Roquejoffre, Jean Michel; Savin, Ovidiu (2010), "Nonlocal Minimal Surfaces", Communications on Pure and Applied Mathematics 63 (9): 1111–1144, doi:10.1002/cpa.20331, ISSN 0003-486X, http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract 
  2. 2.0 2.1 2.2 Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 
  3. 3.0 3.1 Guillen, N.; Schwab, R. (2010), "Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations", Arxiv preprint arXiv:1101.0279  Cite error: Invalid <ref> tag; name "GS" defined multiple times with different content
  4. Barles, Guy; Imbert, Cyril (2008), "Second-order elliptic integro-differential equations: viscosity solutions' theory revisited", Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 25 (3): 567–585, doi:10.1016/j.anihpc.2007.02.007, ISSN 0294-1449, http://dx.doi.org/10.1016/j.anihpc.2007.02.007 
  5. Silvestre, Luis (To appear), "Holder estimates for advection fractional-diffusion equations", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze 
  6. Kassmann, Moritz (2009), "A priori estimates for integro-differential operators with measurable kernels", Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6 
  7. 7.0 7.1 Caffarelli, Luis A.; Vasseur, Alexis (2010), "Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation", Annals of Mathematics. Second Series 171 (3): 1903–1930, doi:10.4007/annals.2010.171.1903, ISSN 0003-486X, http://dx.doi.org/10.4007/annals.2010.171.1903 
  8. Friedlander, S.; Vicol, V. (2011), "Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics", Annales de l'Institut Henri Poincare (C) Non Linear Analysis 
  9. Seregin, G.; Silvestre, Luis; Sverak, V.; Zlatos, A. (2010), "On divergence-free drifts", Arxiv preprint arXiv:1010.6025 
  10. Caffarelli, Luis A.; Salsa, Sandro; Silvestre, Luis (2008), "Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian", Inventiones Mathematicae 171 (2): 425–461, doi:10.1007/s00222-007-0086-6, ISSN 0020-9910, http://dx.doi.org/10.1007/s00222-007-0086-6 
  11. Petrosyan, A.; Garofalo, N. (2009), "Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem", Inventiones Mathematicae (Berlin, New York: Springer-Verlag) 177 (2): 415–461, ISSN 0020-9910