Drift-diffusion equations and Open problems: Difference between pages

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A drift-(fractional)diffusion equation refers to an evolution equation of the form
= Problems for integro-differential equations with rough coefficients or nonlinear equations =
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
== Hölder estimates for singular integro-differential equations ==
where $b$ is any vector field. The stationary version can also be of interest
\[ b \cdot \nabla u + (-\Delta)^s u = 0.\]


This type of equations appear 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.
Consider an integro-differential equation of the form
\[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]
(An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin)


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.
[[Hölder estimates]] are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy.
\[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \]
for all radius $R>0$ and $x \in B_1$, for some given constant $\alpha \in (0,2)$. This is the sharp assumption for stable operators that are independent of $x$ <ref name="ros2014regularity" />.


== Scaling ==
[[Hölder estimates]] are not known to hold under such generality. For the current methods, singular measures $\mu_x$ (without an absolutely continuous part) are out of reach. A new idea is needed in order to solve this problem.


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
Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.
\[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]


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.
== An integral ABP estimate ==


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.
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?


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.
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?


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


=== Kato classes ===
== Holder estimates for parabolic equations with variable order ==
The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller, effectively making the drift term a small perturbation of the fractional diffusion. The assumption is tailor made in order to make [[bootstrapping]] arguments possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation <ref name="BJ"/> <ref name="CKS"/>.


=== $C^{1,\alpha}$ estimates ===
[[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 <ref name="BK"/> <ref name="S" />. Such estimate is not available for parabolic equations and it is not clear whether it holds.
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.


The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result has been proved for the case $s \leq 1/2$ <ref name="S3"/>, and it has also been proved for a stationary eigenvalue problem if $s \geq 1/2$ <ref name="P"/>.
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?


== Scale invariant results ==
== 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?


=== Divergence-free vector fields ===
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.
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.
\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 $x$?


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"/>.
Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.<ref name="BI"/>


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


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


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.
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 $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).


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 $\mathrm{div} \ b =0$.  
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 $p$.
 
= Open problems for equations related to fluids =
 
== Well posedness of the supercritical [[surface quasi-geostrophic equation]] and related problems ==
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 + (-\Delta)^s \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 that the solution of this problem should be preceded by a better understanding of the inviscid problem (with the fractional diffusion term removed).
 
== Well posedness of the Hilbert flow problem ==
 
The Hilbert flow problem is a simple 1D toy model for fluid equations in higher dimensions. It was originally suggested in a paper by Cordoba, Cordoba and Fontelos.<ref name="cordobacordoba2005" /> The equation is in terms of a scalar function $\theta(t,x)$. Here $x \in \R$ is a one dimensional variable.
\[ \theta_t + \mathrm H\theta \, \theta_x = 0.\]
The operator $\mathrm H$ stands for the Hilbert transform. There are several independent proofs that this equation develops singularities in finite time.<ref name="cordobacordoba2005" />
<ref name="CCF2"/> <ref name="HDong"/> <ref name="K"/> <ref name="SV" /> The equation still develops singularities in finite time if we add fractional diffusion
\[ \theta_t + \mathrm H\theta \, \theta_x + (-\Delta)^s \theta = 0,\]
provided that $s < 1/4$.<ref name="HDong"/> <ref name="SV"/> <ref name="K"/> <ref name="li2011one" /> The equation is known to be classically well posed for $s \geq 1/2$. In the range $s \in [1/4,1/2)$, it is not known whether singularities may occur in finite time.
 
Silvestre and Vicol conjectured that the solution $\theta$ satisfies an a priori estimate in $C^{1/2}$ for positive time, both in the viscous and inviscid model.<ref name="SV" /> If this conjecture turns out to be true, the equation above will be well posed when $s > 1/4$.
 
= Open problems related to minimal surfaces and free boundaries =
 
== Regularity of [[nonlocal minimal surfaces]] ==
 
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 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 $n < 8$.
 
== 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 <ref name="CRS16" />.
 
== 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 <ref name="CSS"/>. 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 <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]] 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''?


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
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<ref name="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
<ref name="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
<ref name="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="SSSZ">{{Citation | last1=Seregin | first1=G. | last2=Silvestre | first2=Luis | last3=Sverak | first3=V. | last4=Zlatos | first4=A. | title=On divergence-free drifts | year=2010 | journal=Arxiv preprint arXiv:1010.6025}}</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="FV">{{Citation | last1=Friedlander | first1=S. | last2=Vicol | first2=V. | title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics | year=2011 | journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</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="CRS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Roquejoffre | first2=Jean Michel |last3= Savin | first3= Ovidiu | title= Nonlocal Minimal Surfaces | url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract | doi=10.1002/cpa.20331 | year=2010 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0003-486X | volume=63 | issue=9 | pages=1111–1144}}</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>
 
<ref name="S3">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to an equation with drift and fractional diffusion. | year=To appear | journal=Indiana University Mathematical Journal}}</ref>
<ref name="CRS16">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Ros-Oton | first2=Xavier |last3= Serra | first3= Joaquim | title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries | year=2016 | journal=[[preprint arXiv (2016)]]}}</ref>
<ref name="P">{{Citation | last1=Priola | first1=E. | title=Pathwise uniqueness for singular SDEs driven by stable processes | year=2010 | journal=Arxiv preprint arXiv:1005.4237}}</ref>
 
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<ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref>
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<ref name="CKS">{{Citation | last1=Chen | first1=Z.Q. | last2=Kim | first2=P. | last3=Song | first3=R. | title=Dirichlet heat kernel estimates for fractional Laplacian under gradient perturbation | year=To appear | journal=Annals of Probability}}</ref>
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<ref name="cordobacordoba2005">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Formation of singularities for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.4007/annals.2005.162.1377 | journal=Ann. of Math. (2) | issn=0003-486X | year=2005 | volume=162 | pages=1377--1389 | doi=10.4007/annals.2005.162.1377}}</ref>
<ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref>
<ref name="HDong">{{Citation | last1=Dong | first1= Hongjie | title=Well-posedness for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 | journal=J. Funct. Anal. | issn=0022-1236 | year=2008 | volume=255 | pages=3070--3097 | doi=10.1016/j.jfa.2008.08.005}}</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 | journal=Math. Model. Nat. Phenom. | issn=0973-5348 | year=2010 | volume=5 | pages=225--255 | doi=10.1051/mmnp/20105410}}</ref>
<ref name="CCF2">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation | url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 | journal=J. Math. Pures Appl. (9) | issn=0021-7824 | year=2006 | volume=86 | pages=529--540 | doi=10.1016/j.matpur.2006.08.002}}</ref>
}}
}}
[[Category:Semilinear equations]]

Revision as of 15:45, 25 January 2016

Problems for integro-differential equations with rough coefficients or nonlinear equations

Hölder estimates for singular integro-differential equations

Consider an integro-differential equation of the form \[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\] (An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin)

Hölder estimates are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy. \[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \] for all radius $R>0$ and $x \in B_1$, for some given constant $\alpha \in (0,2)$. This is the sharp assumption for stable operators that are independent of $x$ [1].

Hölder estimates are not known to hold under such generality. For the current methods, singular measures $\mu_x$ (without an absolutely continuous part) are out of reach. A new idea is needed in order to solve this problem.

Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.

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]

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 [4] [5]. 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?

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.[6]

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 $p$.

Open problems for equations related to fluids

Well posedness of the supercritical surface quasi-geostrophic equation and related problems

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 + (-\Delta)^s \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 that the solution of this problem should be preceded by a better understanding of the inviscid problem (with the fractional diffusion term removed).

Well posedness of the Hilbert flow problem

The Hilbert flow problem is a simple 1D toy model for fluid equations in higher dimensions. It was originally suggested in a paper by Cordoba, Cordoba and Fontelos.[10] The equation is in terms of a scalar function $\theta(t,x)$. Here $x \in \R$ is a one dimensional variable. \[ \theta_t + \mathrm H\theta \, \theta_x = 0.\] The operator $\mathrm H$ stands for the Hilbert transform. There are several independent proofs that this equation develops singularities in finite time.[10] [11] [12] [13] [14] The equation still develops singularities in finite time if we add fractional diffusion \[ \theta_t + \mathrm H\theta \, \theta_x + (-\Delta)^s \theta = 0,\] provided that $s < 1/4$.[12] [14] [13] [15] The equation is known to be classically well posed for $s \geq 1/2$. In the range $s \in [1/4,1/2)$, it is not known whether singularities may occur in finite time.

Silvestre and Vicol conjectured that the solution $\theta$ satisfies an a priori estimate in $C^{1/2}$ for positive time, both in the viscous and inviscid model.[14] If this conjecture turns out to be true, the equation above will be well posed when $s > 1/4$.

Open problems related to minimal surfaces and free boundaries

Regularity of nonlocal minimal surfaces

A nonlocal minimal surface that is sufficiently flat is known to be smooth [16]. 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 $n < 8$.

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 [17].

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 [18]. 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 [19]. 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 because of the decay at infinity requirement.
  2. In case that a point of a third category exist, is the free boundary smooth around these points in the third category?

References

  1. Ros-Oton, Xavier; Serra, Joaquim, "Regularity theory for general stable operators", arXiv preprint arXiv:1412.3892 
  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. 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 
  5. 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 
  6. 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 
  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. 10.0 10.1 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 
  11. Córdoba, Antonio; Córdoba, Diego; Fontelos, Marco A. (2006), "Integral inequalities for the Hilbert transform applied to a nonlocal transport equation", J. Math. Pures Appl. (9) 86: 529--540, doi:10.1016/j.matpur.2006.08.002, ISSN 0021-7824, http://dx.doi.org/10.1016/j.matpur.2006.08.002 
  12. 12.0 12.1 Dong, Hongjie (2008), "Well-posedness for a transport equation with nonlocal velocity", J. Funct. Anal. 255: 3070--3097, doi:10.1016/j.jfa.2008.08.005, ISSN 0022-1236, http://dx.doi.org/10.1016/j.jfa.2008.08.005 
  13. 13.0 13.1 Kiselev, A. (2010), "Regularity and blow up for active scalars", Math. Model. Nat. Phenom. 5: 225--255, doi:10.1051/mmnp/20105410, ISSN 0973-5348, http://dx.doi.org/10.1051/mmnp/20105410 
  14. 14.0 14.1 14.2 Silvestre, Luis; Vicol, Vlad, "On a transport equation with nonlocal drift", arXiv preprint arXiv:1408.1056 
  15. Li, Dong; Rodrigo, José L (2011), "On a one-dimensional nonlocal flux with fractional dissipation", SIAM Journal on Mathematical Analysis 43: 507--526 
  16. 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 
  17. Caffarelli, Luis A.; Ros-Oton, Xavier; Serra, Joaquim (2016), "Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries", preprint arXiv (2016) 
  18. 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 
  19. 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