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Hölder continuity of the solutions can sometimes be proved only from ellipticity
An equation is called semilinear if it consists of the sum of a well understood linear term plus a lower order nonlinear term. For elliptic and parabolic equations, the two effective possibilities for the linear term is to be either the [[fractional Laplacian]] or the [[fractional heat equation]].
assumptions on the equation, without depending on smoothness of the
coefficients. This allows great flexibility in terms of applications of the
result. The corresponding result for elliptic equations of second order is the
[[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]]. In most cases, one can deduce the Hölder estimates from the Harnack inequality. However, there are simple example of integro-differential equations for which the Hölder estimates hold and the Harnack inequality does not <ref name="rang2013h" /> <ref name="bogdan2005harnack" />.
Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form
\[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \]
can be thought of as semilinear equations even if $s<1/2$.


There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
== Some common semilinear equations ==
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
=== The most common elliptic equation in the world (provisional title) ===
$L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.
(small). It is very important when an estimate allows for a very rough dependence of
\[ (-\Delta)^s u = f(u). \]
$L_x$ with respect to $x$, since the result then applies to the linearization of
If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].
(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 applicable.


In the non variational setting the integro-differential operators $L_x$ are
Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.
assumed to belong to some family, but no continuity is assumed for its
dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
K(x,y) \, dy$$
Within the context of nonlocal equations, we would be interested on a regularization effect caused by the integral term and not the second order part of the equation. Because o that, the coefficients $a_{ij}(x)$ are usually zero.


Since [[linear integro-differential operators]] allow for a great flexibility of
=== Reaction diffusion equations ===
equations, there are several variations on the result: different assumptions on
This general class refers to the equations we get by adding a zeroth order term to the right hand side of a heat equation. For the fractional case, it would look like
the kernels, mixed local terms, evolution equations, etc. The linear equation
\[ u_t + (-\Delta)^s u = f(u). \]
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
The case $f(u) = u(1-u)$ corresponds to the KPP/Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior <ref name="CR"/>, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.
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 ==
=== Burgers equation with fractional diffusion ===
It refers to the parabolic equation for a function on the real line $u:[0,+\infty) \times \R \to \R$,
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
The equation is known to be well posed if $s \geq 1/2$ and to develop shocks if $s<1/2$ <ref name="KNS"/>. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times<ref name="CCS"/><ref name="K"/>.


=== Elliptic form ===
=== [[Surface quasi-geostrophic 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
It refers to the parabolic equation for a scalar function on the plane $\theta:[0,+\infty) \times \R^2 \to \R$,
\[
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]
L(u,x) = 0 \ \ \text{in } B_1,
where $u = R^\perp \theta$ (and $R$ is the Riesz transform).
\]
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.
The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation.


For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
=== Conservation laws with fractional diffusion ===
\begin{align*}
(aka "fractal conservation laws")
M^+u \geq 0 \ \ \text{in } B_1, \\
It refers to parabolic equations of the form
M^-u \leq 0 \ \ \text{in } B_1.
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
\end{align*}
The Cauchy problem is known to be well posed classically if $s > 1/2$ <ref name="DI"/>. For $s<1/2$ there are viscosity solutions that are not $C^1$.
where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.


=== Parabolic form ===
The critical case $s=1/2$ appears not to be written anywhere. However, it can be solved following the same method as for the Hamilton-Jacobi equations with fractional diffusion (below) <ref name="S"/> or the modulus of continuity approach <ref name="K"/>.
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
\[
u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]


=== Hamilton-Jacobi equation with fractional diffusion ===
It refers to the parabolic equation
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]


== Estimates which blow up as the order goes to two ==
The Cauchy problem is known to be well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.
 
=== Non variational case ===
 
The Hölder estimates were first obtained using probabilistic techniques <ref
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 needs to be
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 $L^\infty$.
 
A particular case in which this result applies is the uniformly elliptic case.
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
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
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
$$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
which should be understood in the sense of distributions.
 
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" />.


The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]] <ref name="DI"/>. The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]] <ref name="S"/>.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
<ref name="KNS">{{Citation | last1=Kiselev | first1=Alexander | last2=Nazarov | first2=Fedor | last3=Shterenberg | first3=Roman | title=Blow up and regularity for fractal Burgers equation | year=2008 | journal=Dynamics of Partial Differential Equations | issn=1548-159X | volume=5 | issue=3 | pages=211–240}}</ref>
estimates for solutions of integro-differential equations like the fractional
<ref name="S">{{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>
Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
<ref name="CCS">{{Citation | last1=Chan | first1=Chi Hin | last2=Czubak | first2=Magdalena | last3=Silvestre | first3=Luis | title=Eventual regularization of the slightly supercritical fractional Burgers equation | url=http://dx.doi.org/10.3934/dcds.2010.27.847 | doi=10.3934/dcds.2010.27.847 | year=2010 | journal=Discrete and Continuous Dynamical Systems. Series A | issn=1078-0947 | volume=27 | issue=2 | pages=847–861}}</ref>
doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
<ref name="K">{{Citation | last1=Kiselev | first1=A. | title=Nonlocal maximum principles for active scalars | year=to appear | journal=Advances in Mathematics}}</ref>
Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
<ref name="DI">{{Citation | last1=Droniou | first1=Jérôme | last2=Imbert | first2=Cyril | title=Fractal first-order partial differential equations | url=http://dx.doi.org/10.1007/s00205-006-0429-2 | doi=10.1007/s00205-006-0429-2 | year=2006 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | volume=182 | issue=2 | pages=299–331}}</ref>
pages=1155–1174}}</ref>
<ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
<ref name="CS">{{Citation | last1=Cabre | first1=X. | last2=Sire | first2=Yannick | title=Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates | year=2010 | journal=Arxiv preprint arXiv:1012.0867}}</ref>
first2=Luis | title=Regularity theory for fully nonlinear integro-differential
<ref name="CC"> {{Citation | last1=Cabré | first1=Xavier | last2=Cinti | first2=E. | title=Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian | year=2010 | journal=Discrete and Continuous Dynamical Systems (DCDS-A) | volume=28 | issue=3 | pages=1179–1206}} </ref>
equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 |
<ref name="LF">{{Citation | last1=Frank | first1=R.L. | last2=Lenzmann | first2=E. | title=Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$ | year=2010 | journal=Arxiv preprint arXiv:1009.4042}}</ref>
year=2009 | journal=[[Communications on Pure and Applied Mathematics]] |
<ref name="FQT">{{Citation | last1=Felmer | first1=P. | last2=Quaas | first2=A. | last3=Tan | first3=J. | title=Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian.}}</ref>
issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="SV"> {{Citation | last1=Sire | first1=Yannick | last2=Valdinoci | first2=E. | title=Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result | publisher=[[Elsevier]] | year=2009 | journal=Journal of Functional Analysis | issn=0022-1236 | volume=256 | issue=6 | pages=1842–1864}} </ref>
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
<ref name="PSV">{{Citation | last1=Palatucci | first1=G. | last2=Valdinoci | first2=E. | last3=Savin | first3=O. | title=Local and global minimizers for a variational energy involving a fractional norm | year=2011 | journal=Arxiv preprint arXiv:1104.1725}}</ref>
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
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| 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
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pages=1249–1259}}</ref>
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pages=375–388}}</ref>
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}}
}}

Revision as of 10:51, 2 July 2011

An equation is called semilinear if it consists of the sum of a well understood linear term plus a lower order nonlinear term. For elliptic and parabolic equations, the two effective possibilities for the linear term is to be either the fractional Laplacian or the fractional heat equation.

Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form \[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \] can be thought of as semilinear equations even if $s<1/2$.

Some common semilinear equations

The most common elliptic equation in the world (provisional title)

Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs. \[ (-\Delta)^s u = f(u). \] If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard bootstrapping.

Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained [1] [2] [3] [4] [5] [6].

Reaction diffusion equations

This general class refers to the equations we get by adding a zeroth order term to the right hand side of a heat equation. For the fractional case, it would look like \[ u_t + (-\Delta)^s u = f(u). \]

The case $f(u) = u(1-u)$ corresponds to the KPP/Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior [7], etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.

Burgers equation with fractional diffusion

It refers to the parabolic equation for a function on the real line $u:[0,+\infty) \times \R \to \R$, \[ u_t + u \ u_x + (-\Delta)^s u = 0 \] The equation is known to be well posed if $s \geq 1/2$ and to develop shocks if $s<1/2$ [8]. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times[9][10].

Surface quasi-geostrophic equation

It refers to the parabolic equation for a scalar function on the plane $\theta:[0,+\infty) \times \R^2 \to \R$, \[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \] where $u = R^\perp \theta$ (and $R$ is the Riesz transform).

The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation.

Conservation laws with fractional diffusion

(aka "fractal conservation laws") It refers to parabolic equations of the form \[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\] The Cauchy problem is known to be well posed classically if $s > 1/2$ [11]. For $s<1/2$ there are viscosity solutions that are not $C^1$.

The critical case $s=1/2$ appears not to be written anywhere. However, it can be solved following the same method as for the Hamilton-Jacobi equations with fractional diffusion (below) [12] or the modulus of continuity approach [10].

Hamilton-Jacobi equation with fractional diffusion

It refers to the parabolic equation \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]

The Cauchy problem is known to be well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.

The subcritical case $s>1/2$ can be solved with classical bootstrapping [11]. The critical case $s=1/2$ was solved using the regularity results for drift-diffusion equations [12].

References

  1. ↑ Cabre, X.; Sire, Yannick (2010), "Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates", Arxiv preprint arXiv:1012.0867 
  2. ↑ Cabré, Xavier; Cinti, E. (2010), "Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian", Discrete and Continuous Dynamical Systems (DCDS-A) 28 (3): 1179–1206 
  3. ↑ Frank, R.L.; Lenzmann, E. (2010), "Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$", Arxiv preprint arXiv:1009.4042 
  4. ↑ Felmer, P.; Quaas, A.; Tan, J., Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian. 
  5. ↑ Sire, Yannick; Valdinoci, E. (2009), "Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result", Journal of Functional Analysis (Elsevier) 256 (6): 1842–1864, ISSN 0022-1236 
  6. ↑ Palatucci, G.; Valdinoci, E.; Savin, O. (2011), "Local and global minimizers for a variational energy involving a fractional norm", Arxiv preprint arXiv:1104.1725 
  7. ↑ Cabré, Xavier; Roquejoffre, Jean-Michel (2009), "Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire", Comptes Rendus Mathématique. Académie des Sciences. Paris 347 (23): 1361–1366, doi:10.1016/j.crma.2009.10.012, ISSN 1631-073X, http://dx.doi.org/10.1016/j.crma.2009.10.012 
  8. ↑ Kiselev, Alexander; Nazarov, Fedor; Shterenberg, Roman (2008), "Blow up and regularity for fractal Burgers equation", Dynamics of Partial Differential Equations 5 (3): 211–240, ISSN 1548-159X 
  9. ↑ Chan, Chi Hin; Czubak, Magdalena; Silvestre, Luis (2010), "Eventual regularization of the slightly supercritical fractional Burgers equation", Discrete and Continuous Dynamical Systems. Series A 27 (2): 847–861, doi:10.3934/dcds.2010.27.847, ISSN 1078-0947, http://dx.doi.org/10.3934/dcds.2010.27.847 
  10. ↑ 10.0 10.1 Kiselev, A. (to appear), "Nonlocal maximum principles for active scalars", Advances in Mathematics 
  11. ↑ 11.0 11.1 Droniou, Jérôme; Imbert, Cyril (2006), "Fractal first-order partial differential equations", Archive for Rational Mechanics and Analysis 182 (2): 299–331, doi:10.1007/s00205-006-0429-2, ISSN 0003-9527, http://dx.doi.org/10.1007/s00205-006-0429-2 
  12. ↑ 12.0 12.1 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 
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