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| A drift-(fractional)diffusion equation refers to an evolution equation of the form
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| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
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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 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.
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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.
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| == Scaling ==
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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
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| \[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]
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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.
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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.
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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.
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| == Pertubative results ==
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| === Kato classes ===
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| The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller making the drift term a small perturbation of the fractional diffusion. The precise assumption varies from article to article and they are usually tailor made in order to make some [[bootstrapping]] argument 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"/>.
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| === $C^{1,\alpha}$ estimates ===
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| 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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| 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 holds both if $s > 1/2$ <ref name="P"/> as if $s \leq 1/2$ <ref name="S3"/>.
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| == Scale invariant results ==
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| === Divergence-free vector fields ===
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| 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"/>.
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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"/>.
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| === Vector fields with arbitrary divergence ===
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| 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.
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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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| <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>
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| <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="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>
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| <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>
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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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| [[Category:Semilinear equations]]
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