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| A drift-(fractional)diffusion equation refers to an evolution equation of the form
| | In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that are minimizers or critical points (within a class of given admissible configurations) of the energy functional: |
| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
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| where $b$ is any vector fields. The stationary version can also be of interest
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| \[ b \cdot \nabla u + (-\Delta)^s u = 0.\]
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| 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.
| | \[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \] |
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| 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.
| | It can be checked easily that this agrees (save for a factor of $2$) with norm of the characteristic function $\chi_E$ in the homogenous Sobolev space $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$. |
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| == Scaling ==
| | Classically, [[minimal surfaces]] (or generally [[surfaces of constant mean curvature]] ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart. |
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| 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
| | Nonlocal minimal surfaces then describe physical phenomena where the interaction potential does not decay fast enough as particles get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials |
| \[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]
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| 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.
| | \[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \] |
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| 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.
| | == Definition == |
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| 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.
| | Following the most accepted convention for [[minimal surfaces]], a (classical) nonlocal minimal surface is (given $s\in (0,1)$) the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\Sigma$ is at least $C^{1,s+\epsilon}$ and more importantly, |
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| == Pertubative results == | | \[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\] |
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| === Kato classes ===
| | In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. The quantity $H_s(x)$ is called the "Nonlocal mean curvature of order $s$ of $\Sigma$ at $x$", or briefly, "Nonlocal mean curvature". |
| The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller. | |
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| === $C^{1,\alpha}$ estimates ===
| | Example: Suppose that $E$ and $\Omega$ are such that for any other set $F$ such that $F \Delta E \subset \subset \Omega$ (i.e. $F$ agrees with $E$ outside $\Omega$) we have |
| 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.
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| == Scale invariant results ==
| | \[J_s(E) \leq J_s(F) \] |
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| === Divergence-free vector fields ===
| | Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$. |
| 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.
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| 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"/>.
| | <div style="background:#DDEEFF;"> |
| | <blockquote> |
| | '''Note''' For this definition to make sense, $\Sigma$ must be the boundary of some open set $E$, in this article, we will often refer to the set $E$ itself as "the" minimal surface, and no confusion should arise from this. |
| | </blockquote> |
| | </div> |
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| 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"/>.
| | == Nonlocal mean curvature == |
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| === Vector fields with arbitrary divergence === | | == Surfaces minimizing non-local energy functionals == |
| 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.
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| 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 Caffarelli-Roquejoffre-Savin Regularity Theorem== |
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| 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 $\mathrm{div} \ b =0$.
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| == References ==
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| {{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>
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| <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>
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| <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>
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| <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>
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| <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>
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| }}
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In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that are minimizers or critical points (within a class of given admissible configurations) of the energy functional:
\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]
It can be checked easily that this agrees (save for a factor of $2$) with norm of the characteristic function $\chi_E$ in the homogenous Sobolev space $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$.
Classically, minimal surfaces (or generally surfaces of constant mean curvature ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart.
Nonlocal minimal surfaces then describe physical phenomena where the interaction potential does not decay fast enough as particles get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials
\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
Definition
Following the most accepted convention for minimal surfaces, a (classical) nonlocal minimal surface is (given $s\in (0,1)$) the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\Sigma$ is at least $C^{1,s+\epsilon}$ and more importantly,
\[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\]
In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. The quantity $H_s(x)$ is called the "Nonlocal mean curvature of order $s$ of $\Sigma$ at $x$", or briefly, "Nonlocal mean curvature".
Example: Suppose that $E$ and $\Omega$ are such that for any other set $F$ such that $F \Delta E \subset \subset \Omega$ (i.e. $F$ agrees with $E$ outside $\Omega$) we have
\[J_s(E) \leq J_s(F) \]
Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$.
Note For this definition to make sense, $\Sigma$ must be the boundary of some open set $E$, in this article, we will often refer to the set $E$ itself as "the" minimal surface, and no confusion should arise from this.
Nonlocal mean curvature
Surfaces minimizing non-local energy functionals
The Caffarelli-Roquejoffre-Savin Regularity Theorem