# Open problems

### From Mwiki

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 $s \geq 1/2$.

There are difficult open problems related to simpler active scalar equations as well. The *Hilbert flow problem* refers to the equation
\[ \theta_t + H\theta \, \theta_x + (-\Delta)^s \theta = 0.\]
Here $\theta(t,x)$ is a function of $t \in [0,\infty)$ and $x \in \R$. The equation is known to be well posed for $s \geq 1/2$ and it is known to develop singularities in finite time when $s < 1/4$. For $s$ in the interval $s \in [1/4,1/2)$, it is not known whether singularities in finite time may occur.^{[1]}^{[2]}^{[3]}

## Regularity of nonlocal minimal surfaces

A nonlocal minimal surface that is sufficiently flat is known to be smooth ^{[4]}. 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

- $A$ 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

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

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 $L^\infty$ ^{[5]} (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels ^{[6]}. 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 $K$.^{[6]}

## 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.^{[7]}

## A local $C^{s+\alpha}$ estimate for the the Pucci equation of order $s$

Assume that $u : \R^n \to \R$ is a bounded function satisfying the equation \[ M^+ u = 0 \qquad \text{in } B_1.\] Here $M^+$ is the Monster Pucci operator of order $s$. Is it true that $u \in C^{s+\alpha}(B_{1/2})$?

In the proof of the Nonlocal Evans-Krylov theorem ^{[8]}, it is assumed that the second derivatives of the kernels satisfy certain bounds. Because of this, it cannot be applied to this case.

UPDATE: this question was resolved by Joaquim Serra ^{[9]}.

## 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 and radially symmetric. 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.

UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra ^{[10]}.

## 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 $b \in L^\infty(BMO)$ ^{[11]}. 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) ^{[12]} ^{[13]}. 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 ^{[11]} can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.

## Complete understanding 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 ^{[14]}. Other points on the free boundary are classified as singular, and for $s=\frac12$ they are shown to be contained in a lower dimensional differentiable surface, and therefore to be rare ^{[15]}. 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 because of the decay at infinity requirement.
- In case that a 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.

## Holder estimates for parabolic equations with variable order

Holder estimates are known for elliptic and parabolic integro-differential equations with rough kernels. For elliptic equations, these estimates are available even when the order of the equation changes from point to point ^{[16]} ^{[17]}. Such estimate is not available for parabolic equations and it is not clear whether it holds.

More precisely, we would like to study a parabolic equation of the form \[ u_t(t,x) = \int_{\R^n} (u(t,x+y) - u(t,x)) K(t,x,y) dy.\] Here $K$ is symmetric (i.e. $K(t,x,y) = K(t,x,-y)$) and satisfies the bounds \[ \frac \lambda {|y|^{n+s(t,x)}} \leq K(t,x,y) \leq \frac \Lambda {|y|^{n+s(t,x)}}.\] The order of the equation $s(t,x) \in (0,1)$ changes from point to point and it should stay strictly away from zero. It would also make sense to study other families of linear integro-differential operators. Does a parabolic Holder estimate hold in this case?

## References

- ↑ Córdoba, Antonio; Córdoba, Diego; Fontelos, Marco A. (2005), "Formation of singularities for a transport equation with nonlocal velocity",
*Ann. of Math. (2)***162**: 1377--1389, doi:10.4007/annals.2005.162.1377, ISSN 0003-486X, http://dx.doi.org/10.4007/annals.2005.162.1377 - ↑ Li, Dong; Rodrigo, José L (2011), "On a one-dimensional nonlocal flux with fractional dissipation",
*SIAM Journal on Mathematical Analysis***43**: 507--526 - ↑ Silvestre, Luis; Vicol, Vlad, "On a transport equation with nonlocal drift",
*arXiv preprint arXiv:1408.1056* - ↑ 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 - ↑ 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 - ↑
^{6.0}^{6.1}Guillen, N.; Schwab, R. (2010), "Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations",*Arxiv preprint arXiv:1101.0279* - ↑ 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 - ↑ Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations",
*Annals of Mathematics*, ISSN 0003-486X - ↑ Serra, Joaquim, "$C^{\sigma+\alpha}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels",
*arXiv preprint arXiv:1405.0930* - ↑ Caffarelli, Luis A.; Ros-Oton, Xavier; Serra, Joaquim (2016), "Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries",
*preprint arXiv (2016)* - ↑
^{11.0}^{11.1}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 - ↑ 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* - ↑ Seregin, G.; Silvestre, Luis; Sverak, V.; Zlatos, A. (2010), "On divergence-free drifts",
*Arxiv preprint arXiv:1010.6025* - ↑ 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 - ↑ 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 - ↑ Bass, Richard F.; Kassmann, Moritz (2005), [http://dx.doi.org/10.1080/03605300500257677 "Hölder continuity of harmonic functions with respect to
operators of variable order"],
*Communications in Partial*Differential Equations**30**(7): 1249–1259, doi:10.1080/03605300500257677, ISSN 0360-5302, http://dx.doi.org/10.1080/03605300500257677 - ↑ Silvestre, Luis (2006), [http://dx.doi.org/10.1512/iumj.2006.55.2706 "Hölder
estimates for solutions of integro-differential equations like the fractional
Laplace"],
*Indiana University*Mathematics Journal**55**(3): 1155–1174, doi:10.1512/iumj.2006.55.2706, ISSN 0022-2518, http://dx.doi.org/10.1512/iumj.2006.55.2706