Drift-diffusion equations and Nonlocal minimal surfaces: Difference between pages

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A drift-(fractional)diffusion equation refers to an evolution equation of the form
Broadly speaking, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) the energy functional:
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
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.
\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]


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 $E$.  


== 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.


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 are apart, so that two particles on different phases and far from the interface still contribute a non-trivial amount to the total interaction energy, in particular, one may consider much more general energy functionals corresponding to different interaction potentials
\[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]


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.
\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]


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.
== Nonlocal mean curvature ==


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.
== Surfaces minimizing non-local energy functionals ==


== Pertubative results ==
== Regularity results ==
 
=== Kato classes ===
The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller, 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 ===
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"/>.
 
== Scale invariant results ==
 
=== Divergence-free vector fields ===
If the vector field $b$ is divergence free, the method of [[De Giorgi-Nash-Moser]] can be adapted to show that the solution $u$ becomes immediately Holder continuous. This is a variation of the [[Holder estimates]] for integro-differential equations in variational form.
 
In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(BMO)$ <ref name="CV"/> <ref name="KN"/>.
 
In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ <ref name="CW"/>.
 
=== Vector fields with arbitrary divergence ===
For any bounded vector field $b$, one method for obtaining [[Holder estimates]] for integro-differential equations can be used to show that $u$ becomes immediately Holder continuous.
 
In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(L^\infty)$ <ref name="S1"/>, which is a marginally stronger assumption than the one needed if $b$ was assumed divergence-free.
 
In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ <ref name="S2"/>, which is the same assumption than the one needed if $b$ was assumed divergence-free. Therefore, if $s<1/2$, there is no known advantage in assuming $\mathrm{div} \ b =0$.
 
== References ==
{{reflist|refs=
<ref name="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
<ref name="KN">{{Citation | last1=Kiselev | first1=A. | last2=Nazarov | first2=F. | title=A variation on a theme of Caffarelli and Vasseur | year=2009 | journal=Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u\i Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI) | issn=0373-2703 | volume=370 | pages=58–72}}</ref>
<ref name="CW">{{Citation | last1=Constantin | first1=Peter | last2=Wu | first2=Jiahong | title=Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations | url=http://dx.doi.org/10.1016/j.anihpc.2007.10.002 | doi=10.1016/j.anihpc.2007.10.002 | year=2009 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=26 | issue=1 | pages=159–180}}</ref>
<ref name="S1">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion | url=http://dx.doi.org/10.1016/j.aim.2010.09.007 | doi=10.1016/j.aim.2010.09.007 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=2020–2039}}</ref>
<ref name="S2">{{Citation | last1=Silvestre | first1=Luis | title=Holder estimates for advection fractional-diffusion equations | year=To appear | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze}}</ref>
<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="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>
<ref name="S2">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to an equation with drift and fractional diffusion | year=2010 | journal=Arxiv preprint arXiv:1012.2401}}</ref>
<ref name="BJ">{{Citation | last1=Bogdan | first1=Krzysztof | last2=Jakubowski | first2=Tomasz | title=Estimates of heat kernel of fractional Laplacian perturbed by gradient operators | url=http://dx.doi.org/10.1007/s00220-006-0178-y | doi=10.1007/s00220-006-0178-y | year=2007 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=271 | issue=1 | pages=179–198}}</ref>
<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>
}}
 
[[Category:Semilinear equations]]

Revision as of 12:49, 31 May 2011

Broadly speaking, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) 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 are apart, so that two particles on different phases and far from the interface still contribute a non-trivial amount to the total interaction energy, in particular, one may consider much more general energy functionals corresponding to different interaction potentials

\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]

Nonlocal mean curvature

Surfaces minimizing non-local energy functionals

Regularity results