Fully nonlinear integro-differential equations and Drift-diffusion equations: Difference between pages

Revision as of 00:13, 31 May 2011

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 fields. The stationary version can also be of interest \[ b \cdot \nabla u + (-\Delta)^s u = 0.\]

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.

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.

$C^{1,\alpha}$ 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.

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 $L^\infty(BMO)$ ^[1] ^[2].

In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ ^[3].

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 $L^\infty(L^\infty)$ ^[4], 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 $L^\infty(C^{1-2s})$ ^[5], 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

↑ 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
↑ 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
↑ 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
↑ 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
↑ Silvestre, Luis (To appear), "Holder estimates for advection fractional-diffusion equations", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze

[CV-1] 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

[KN-2] 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

[CW-3] 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

[S1-4] 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

[S2-5] Silvestre, Luis (To appear), "Holder estimates for advection fractional-diffusion equations", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze

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[2]

[3]

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[5]

@@ Line 1: / Line 1: @@
-Fully nonlinear integro-differential equations are a nonlocal version of fully nonlinear elliptic equations of the form $F(D^2 u, Du, u, x)=0$. The main examples are the integro-differential [[Bellman equation]] from optimal control, and the [[Isaacs equation]] from stochastic games.
+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 fields. The stationary version can also be of interest
+\[ b \cdot \nabla u + (-\Delta)^s u = 0.\]
-Equations of this type commonly satisfy a [[comparison principle]] and have some [[Regularity results for fully nonlinear integro-differential equations|regularity results]].
+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.
-The general definition of ellipticity provided below does not require a specific form of the equation. However, the main two applications are the two above.
+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.
-== Abstract definition <ref name="CS"/><ref name="CS2"/> ==
+== Scaling ==
-A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement to call $I$ '''elliptic''' is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then
+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
-\[ Iu(x) \leq Iv(x).\]
+\[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]
-We now proceed to define the concept of uniform ellipticity. Given the richness of variations of nonlocal equations, we provide a flexible definition of uniform elliticity depending an arbitrary family of linear operators.
+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.
-Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
+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.
-\begin{align*}
-M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
-M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
-\end{align*}
-We define a nonlinear operator $I$ to be '''uniformly elliptic''' in a domain $\Omega$ with respect to the class $\mathcal{L}$ if it assigns a continuous function $Iu$ to every function $u \in L^\infty(\R^n) \cap C^2(\Omega)$, and moreover for any two such functions $u$ and $v$:
+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.
-\[M^-_\mathcal{L} [u-v](x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} [u-v] (x), \]
-for any $x \in \Omega$.
-A fully nonlinear elliptic equation is an equation of the form $Iu=0$ in $\Omega$, for some elliptic operator $I$.
+== Pertubative results ==
-{{note|text= If $\mathcal L$ consists of purely second order operators of the form $\mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators. It is a ''folklore'' statement that then nonlinear operator $I$ elliptic respect to $\mathcal L$ in the sense described above must coincide with a fully nonlinear elliptic operator of the form $Iu = F(D^2u,x)$. However, this proof may have never been written anywhere.
+=== 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.
-{{note|text=Any uniformly elliptic integro-differential equation with respect to some class $\mathcal L$ satisfies the following $\inf-\sup$ representation
+=== $C^{1,\alpha}$ estimates ===
-\[ Iu(x) = \inf_{v \in C^2 \cap L^\infty} \sup_{L \in \mathcal L} \bigg\{ (Iv(x) - Lv(v)) + Lu(x) \bigg\}.\]
+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.
-Therefore, $I$ coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$ (the index $a$ ranges over all functions $v$ and the index $b$ over all linear operators $L$).
-In the case that $I$ is not uniformly elliptic with respect to any given class, an $\inf-\sup$ formula was also obtained when the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>.
+== Scale invariant results ==
-}}
-== Another definition==
+=== Divergence-free vector fields ===
-Another definition which gives a more concrete structure to the equation has been suggested <ref name="BI"/>. It is not clear if both definitions are equivalent, but both include the most important examples and are amenable of approximately the same methods.
+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.
-Given a family of [[linear integro-differential operators]] $L_\alpha$ indexed by a parameter $\alpha$ which ranges in an arbitrary set $A$, a fully nonlinear elliptic equation is an equation of the 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"/>.
-\[ F(D^2 u, Du, u, x, \{L_\alpha\}_\alpha) = 0 \qquad \text{in } \Omega.\]
-Where the function $F(X,p,z,x,\{i_\alpha\}_\alpha)$ is monotone increasing with respect to $X$ and $\{i_\alpha\}$ and monotone decreasing with respect to $z$.
-Note that the family of linear operators $\{L_\alpha\}$ can range in an arbitrarily large set $A$ (it could even be uncountable).
+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"/>.
-{{note|text= In several articles <ref name="BI"/><ref name="BIC2"/><ref name="BIC"/>, fully nonlinear integro-differential equations of the form $F(D^2 u, Du, u, x, Lu)=f(x)$ are analyzed, where $L$ is one fixed [[linear integro-differential operator]]. This is a rigid structure for purely integro-differential equations because such equation (which would not depend on $D^2u$, $Du$ or $u$) would be forced to be linear: $Lu(x) = [F(x,\cdot)^{-1}f(x)]$.
+=== 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.
-On the other hand, the results in these papers apply to the more general definitions of fully nonlinear integro-differential equations as well. The reason for the restriction to one single integro-differential operator instead of a family $\{L_\alpha\}_\alpha$ seems to be taken only for simplicity.
+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.
-}}
-{{note|text= Whenever a general nonlocal operator corresponding to the first definition has an $\inf-\sup$ representation, it also fits the second definition. Any operator in the form of the second definition obviously fits the first definition. Therefore, for all practical purposes, the two definitions are practically equivalent.
+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 $\mathrm{div} \ b =0$.
-}}
-== Examples ==
-The two main examples are the following.
-* The [[Bellman equation]] is the equality
-\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
-where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
-The equation appears naturally in problems of stochastic control with [[Levy processes]].
-The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
-* The [[Isaacs equation]] is the equality
-\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
-where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
-The equation appears naturally in zero sum stochastic games with [[Levy processes]].
-The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.
 == References ==
 {{reflist|refs=
-<ref name="CS2">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity results for nonlocal equations by approximation | publisher=[[Springer-Verlag]] | location=Berlin, New York | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–30}}</ref>
+<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="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</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="BIC">
+<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>
-{{Citation | last1=Barles | first1=Guy | last2=Chasseigne | first2=Emmanuel | last3=Imbert | first3=Cyril | title=Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | doi=10.4171/JEMS/242 | year=2011 | journal=Journal of the European Mathematical Society (JEMS) | issn=1435-9855 | volume=13 | issue=1 | pages=1–26}}</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="BIC2">{{Citation | last1=Barles | first1=G. | last2=Chasseigne | first2=Emmanuel | last3=Imbert | first3=Cyril | title=On the Dirichlet problem for second-order elliptic integro-differential equations | url=http://dx.doi.org/10.1512/iumj.2008.57.3315 | doi=10.1512/iumj.2008.57.3315 | year=2008 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=57 | issue=1 | pages=213–246}}</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="BI">{{Citation | last1=Barles | first1=Guy | last2=Imbert | first2=Cyril | title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited | url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 | doi=10.1016/j.anihpc.2007.02.007 | year=2008 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=25 | issue=3 | pages=567–585}}</ref>
-<ref name="Guillen-Schwab">{{Citation | last1=Guillen | first1= Nestor | last2=Schwab | first2= Russell W | title=Neumann Homogenization via Integro-Differential Operators | journal=arXiv preprint arXiv:1403.1980}}</ref>
 }}