Hölder estimates and Drift-diffusion equations: Difference between pages

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Hölder continuity of the solutions can sometimes be proved only from ellipticity
A drift-(fractional)diffusion equation refers to an evolution equation of the form
assumptions on the equation, without depending on smoothness of the
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
coefficients. This allows great flexibility in terms of applications of the
where $b$ is any vector field. The stationary version can also be of interest
result. The corresponding result for elliptic equations of second order is the
\[ b \cdot \nabla u + (-\Delta)^s u = 0.\]
[[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.


The Hölder estimates are closely related to the [[Harnack inequality]].
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.


There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
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.
and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
stated in terms of Dirichlet forms. The latter is based on comparison
principles.


A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
== Scaling ==
$L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
(small). It is very important to allow for a very rough dependence of
$L_x$ with respect to $x$, since the result then applies to the linearization of
(fully) nonlinear equations without any extra a priori estimate. On the other
hand, the linearization of a [[fully nonlinear integro-differential equation]] (for example the [[Isaacs equation]] or the [[Bellman equation]]) would inherit the initial assumptions regarding for the kernels with
respect to $y$. Therefore, smoothness (or even structural) assumptions for the
kernels with respect to $y$ can be made keeping such result useful.


In the non variational setting the integro-differential operators $L_x$ are
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
assumed to belong to some family, but no continuity is assumed for its
\[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]
dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
K(x,y) \, dy$$
Since [[linear integro-differential operators]] allow for a great flexibility of
equations, there are several variations on the result: different assumptions on
the kernels, mixed local terms, evolution equations, etc. The linear equation
with rough coefficients is equivalent to the function $u$ satisfying two
inequalities for the [[extremal operators]] corresponding to the family of
operators $L$, which stresses the nonlinear character of the estimates.


As with other estimates in this field too, some Hölder estimates blow up as the
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.
order of the equation converges to two, and others pass to the limit. The
blow-up is a matter of the techniques used in the proof. Only estimates which
are robust are a true generalization of either the [[De Giorgi-Nash-Moser theorem]] or
[[Krylov-Safonov theorem]].  


== The general statement ==
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.


=== Elliptic form ===
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.
The general form of the Hölder estimates for an elliptic problem say that if we have an equation which holds in a domain, and the solution is globally bounded, then the solution is Hölder continuous in the interior of the domain. Typically this is stated in the following form: if $u : \R^d \to \R$ solves
\[
L(u,x) = 0 \ \ \text{in } B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]


There is no lack of generality in assuming that $L$ is a '''linear''' integro-differential operator, provided that there is no regularity assumption on its $x$ dependence.
== Pertubative results ==


For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
=== Kato classes ===
\begin{align*}
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"/>.
M^+u \geq 0 \ \ \text{in } B_1, \\
M^-u \leq 0 \ \ \text{in } B_1.
\end{align*}
where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.


=== Parabolic form ===
=== $C^{1,\alpha}$ estimates ===
The general form of the Hölder estimates for a parabolic problem is also an interior regularity statement for solutions of a parabolic equation. Typically this is stated in the following form: if $u : \R^d \times (-1,0] \to \R$ solves
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.
\[
u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]


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 $s \geq 1/2$ <ref name="P"/> as if $s \leq 1/2$ <ref name="S3"/>.


== Estimates which blow up as the order goes to two ==
== Scale invariant results ==


=== Non variational case ===
=== 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.


The Hölder estimates were first obtained using probabilistic techniques <ref
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"/>.
name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref
name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to
a family satisfying certain set of assumptions. No regularity of any kind is
assumed for $K$ with respect to $x$. The assumption for the family of operators
are
# '''Scaling''': If $L$ belongs to the family, then so does its scaled version
$L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which
could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
# '''Nondegeneracy''': If $K$ is the kernel associated to $L$,
$\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\inf_{B_1} K} \leq C_1$ for
some $C_1$ and $\alpha>0$ independent of $K$.


The right hand side $f$ is assumed to belong to $L^\infty$.
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"/>.


A particular case in which this result applies is the uniformly elliptic case.
=== Vector fields with arbitrary divergence ===
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
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.
where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no
continuity of $s$ respect to $x$ is required.
The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$.
However this assumption can be overcome in the following two situations.
* For $s<1$, the symmetry assumption can be removed if the equation does not
contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy =
f(x)$ in $B_1$.
* For $s>1$, the symmetry assumption can be removed if the drift correction term
is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy =
f(x)$ in $B_1$.


The reason for the symmetry assumption, or the modification of the drift
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.
correction term, is that in the original formulation the term $y \cdot \nabla
u(x) \, \chi_{B_1}(y)$ is not scale invariant.
 
=== Variational case ===
 
A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
$E(u,v)$ satisfying
$$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
\, dy $$
on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note
that $K$ can be assumed to be symmetric because the skew-symmetric part
of $K$ would be ignored by the bilinear form.
 
Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
equation
$$ \int_{\R^n} (u(x+y) - 2 u(x) + u(x-y) ) K(x,y) \, dy = 0$$.
 
It is known that the gradient flow of a Dirichlet form (parabolic version of the
result) becomes instantaneously Hölder continuous <ref name="CCV"/>. The method
of the proof builds an integro-differential version of the parabolic De Giorgi
technique that was developed for the study of critical [[surface
quasi-geostrophic equation]].
 
At some point in the original proof of De Giorgi, it is used that the
characteristic functions of a set of positive measure do not belong to $H^1$.
Moreover, a quantitative estimate is required about the measure of
''intermediate'' level sets for $H^1$ functions. In the integro-differential
context, the required statement to carry out the proof would be the same with
the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
even require a non trivial proof for $s$ close to $2$. The difficulty is
bypassed though an argument that takes advantage of the nonlocal character of
the equation, and hence the estimate blows up as the order approaches two.
 
== Estimates which pass to the second order limit ==
 
=== Non variational case ===
 
An integro-differential generalization of [[Krylov-Safonov]] theorem is
available both in the elliptic <ref name="CS"/> and parabolic <ref name="lara2011regularity"/> setting. The assumption on the kernels are
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq
\frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.
 
The right hand side $f$ is assumed to be in $L^\infty$. The constants in the
Hölder estimate do not blow up as $s \to 2$.
 
=== Variational case ===
 
In the stationary case, it is known that minimizers of Dirichlet forms are
Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
nonlocal setting <ref name="K"/>.
 
== Other variants ==
 
* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a very general class of integral equations <ref name="barles2011" />.


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 ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
<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>
estimates for solutions of integro-differential equations like the fractional
<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>
Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
<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>
doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
<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>
Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
<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>
pages=1155–1174}}</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="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
<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>
first2=Luis | title=Regularity theory for fully nonlinear integro-differential
<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>
equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 |
<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>
year=2009 | journal=[[Communications on Pure and Applied Mathematics]] |
<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>
issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the
American Mathematical Society]] | issn=0894-0347 | issue=24 |
pages=849–869}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori
estimates for integro-differential operators with measurable kernels |
url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6
| year=2009 | journal=Calculus of Variations and Partial Differential Equations
| issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
first2=Moritz | title=Hölder continuity of harmonic functions with respect to
operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
pages=1249–1259}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin |
first2=David A. | title=Harnack inequalities for jump processes |
url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 |
year=2002 | journal=Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory, Geometry and
Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
pages=375–388}}</ref>
<ref name="lara2011regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2011 | pages=1--34}}</ref>
<ref name="zacher2013">{{Citation | last1=Zacher | first1= Rico | title=A De Giorgi--Nash type theorem for time fractional diffusion equations | url=http://dx.doi.org/10.1007/s00208-012-0834-9 | journal=Math. Ann. | issn=0025-5831 | year=2013 | volume=356 | pages=99--146 | doi=10.1007/s00208-012-0834-9}}</ref>
<ref name="barles2011">{{Citation | last1=Barles | first1= Guy | last2=Chasseigne | first2= Emmanuel | last3=Imbert | first3= Cyril | title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | journal=J. Eur. Math. Soc. (JEMS) | issn=1435-9855 | year=2011 | volume=13 | pages=1--26 | doi=10.4171/JEMS/242}}</ref>
}}
}}
[[Category:Semilinear equations]]

Revision as of 19:06, 4 March 2012

A drift-(fractional)diffusion equation refers to an evolution equation of the form \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\] 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.

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.

Scaling

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 \[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]

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.

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.

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.

Pertubative 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 [1] [2].

$C^{1,\alpha}$ 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 holds both if $s \geq 1/2$ [3] as if $s \leq 1/2$ [4].

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.

In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(BMO)$ [5] [6].

In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ [7].

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 $L^\infty(L^\infty)$ [8], 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 $L^\infty(C^{1-2s})$ [9], 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

  1. ↑ Bogdan, Krzysztof; Jakubowski, Tomasz (2007), "Estimates of heat kernel of fractional Laplacian perturbed by gradient operators", Communications in Mathematical Physics 271 (1): 179–198, doi:10.1007/s00220-006-0178-y, ISSN 0010-3616, http://dx.doi.org/10.1007/s00220-006-0178-y 
  2. ↑ Chen, Z.Q.; Kim, P.; Song, R. (To appear), "Dirichlet heat kernel estimates for fractional Laplacian under gradient perturbation", Annals of Probability 
  3. ↑ Priola, E. (2010), "Pathwise uniqueness for singular SDEs driven by stable processes", Arxiv preprint arXiv:1005.4237 
  4. ↑ Silvestre, Luis (To appear), "On the differentiability of the solution to an equation with drift and fractional diffusion.", Indiana University Mathematical Journal 
  5. ↑ 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 
  6. ↑ Kiselev, A.; Nazarov, F. (2009), "A variation on a theme of Caffarelli and Vasseur", Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u\i Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI) 370: 58–72, ISSN 0373-2703 
  7. ↑ Constantin, Peter; Wu, Jiahong (2009), "Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations", Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 26 (1): 159–180, doi:10.1016/j.anihpc.2007.10.002, ISSN 0294-1449, http://dx.doi.org/10.1016/j.anihpc.2007.10.002 
  8. ↑ Silvestre, Luis (2011), "On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion", Advances in Mathematics 226 (2): 2020–2039, doi:10.1016/j.aim.2010.09.007, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.09.007 
  9. ↑ Silvestre, Luis (To appear), "Holder estimates for advection fractional-diffusion equations", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze  Cite error: Invalid <ref> tag; name "S2" defined multiple times with different content
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