Bootstrapping and Starting page: Difference between pages

Revision as of 10:01, 28 January 2012

In this wiki we collect several result about nonlocal elliptic and parabolic equations.

If you want to know what a nonlocal equation refers to, a good starting point would be the Intro to nonlocal equations.

The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity.

We keep a list of open problems and also upcoming events.

Why nonlocal equations

All partial differential equations are a limit case of nonlocal equations. One could even go further and boldly say that in nature all equations are nonlocal, and PDEs are a simplification. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are

Optimal control problems with Levy processes give rise to the Bellman equation, or in general any equation derived from jump processes will be some fully nonlinear integro-differential equation.
In financial mathematics it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the obstacle problem.
Nonlocal electrostatics is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.
The denoising algorithms in nonlocal image processing are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the nonlocal mean curvature flow.
The Boltzmann equation models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified kinetic models can be used to derive the fractional heat equation without resorting to stochastic processes.
In conformal geometry, nonlocal curvatures provide a very rich family of conformally invariant quantities.
In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the surface quasi-geostrophic equation.
Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the Nonlocal porous medium equation, the Hamilton-Jacobi equation with fractional diffusion, conservation laws with fractional diffusion, etc...

Existence and uniqueness results

For a variety of nonlinear elliptic and parabolic equations, the existence of viscosity solutions can be obtained using Perron's method. The uniqueness of solutions is a consequence of the comparison principle.

There are some equations for which this general framework does not work, for example the surface quasi-geostrophic equation. One could say that the underlying reason is that the equation is not purely parabolic, but it has one hyperbolic term.

Regularity results

The regularity tools used for nonlocal equations vary depending on the type of equation.

Nonlinear equations

The starting point to study the regularity of solutions to a nonlinear elliptic or parabolic equation are the Holder estimates which hold under very weak assumptions and rough coefficients. They are related to the Harnack inequality.

For some fully nonlinear integro-differential equation with continuous coefficients, we can prove $C^{1,\alpha}$ estimates.

Under certain hypothesis, the nonlocal Bellman equation from optimal stochastic control has classical solutions due to the nonlocal version of Evans-Krylov theorem.

Semilinear equations

There are several interesting models that are semilinear equations. Those equations consists of either the fractional Laplacian or fractional heat equation plus a nonlinear term.

There are challenging regularity questions especially when the Laplacian interacts with gradient terms in Drift-diffusion equations. A simple method that has been successful in proving the well posedness of some semilinear equations with drift terms in the critical case (when both terms have the same scaling properties) is the conserved modulus of continuity approach, often called "nonlocal maximum principle method".

@@ Line 1: / Line 1: @@
-Bootstrapping is one of the simplest methods to prove regularity of a nonlinear equation. The general idea is described below.
+In this wiki we collect several result about nonlocal elliptic and parabolic equations.
-Assume that $u$ is a solution to some nonlinear equation of any kind. By being a solution to the nonlinear equation, it is also a solution to a linear equation whose coefficients depend on $u$. Typically this is either some form of linearization of the equation. If an a priori estimate is known on $u$, then that provides some assumption on the coefficients of the linear equation that $u$ satisfies. The linear equation, in turn, may provide a new regularity estimate for the solution $u$. If this regularity estimate is stronger than the original a priori estimate, then we can start over and repeat the process to obtain better and better regularity estimates.
+If you want to know what a nonlocal equation refers to, a good starting point would be the [[Intro to nonlocal equations]].
-This is the most elementary example of a [[perturbation method]].
+The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity.
-== Examples ==
+We keep a list of [[open problems]] and also [[upcoming events]].
-=== A simple example ===
+== Why nonlocal equations ==
-Imagine that we have a general semilinear equation of the form
+All partial differential equations are a limit case of nonlocal equations. One could even go further and boldly say that in nature all equations are nonlocal, and PDEs are a simplification. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are
-\[ u_t + (-\Delta)^s u = H(u,Du). \]
+* Optimal control problems with [[Levy processes]] give rise to the [[Bellman equation]], or in general any equation derived from jump processes will be some [[fully nonlinear integro-differential equation]].
-Where $H$ is some smooth function and $s \in (1/2,1]$. Assume that a solution $u$ is known to be Lipschitz. Therefore, $u$ coincides with the solution $v$ of the linear equation
+* In [[financial mathematics]] it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the [[obstacle problem]].
-\begin{align*}
+* [[Nonlocal electrostatics]] is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.
-v(0,x) &= u(0,x) \\
+* The denoising algorithms in [[nonlocal image processing]] are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the [[nonlocal mean curvature flow]].
-v_t + (-\Delta)^s v &= H(u,Du).
+* The [[Boltzmann equation]] models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified [[kinetic models]] can be used to derive the [[fractional heat equation]] without resorting to stochastic processes.
-\end{align*}
+* In conformal geometry, nonlocal curvatures provide a very rich family of conformally invariant quantities.
-Since the right hand side $H(u,Du)$ is bounded, then the solution v must be $C^{2s}$ in space. Since $2s > 1$, then we improved our regularity on $u$ (which is the same as $v$). But now $H(u,Du) \in C^{2s-1}$ and therefore $v \in C^{4s-1}$. Continuing the iteration, we obtain that $u \in C^\infty$.
+* In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasi-geostrophic equation]].
+* Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the [[Nonlocal porous medium equation]], the [[Hamilton-Jacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc...
-The above example is particularly simple because the only estimates used are an assumption that $u$ is Lipschitz and then only the classical estimates for the fractional heat equation. This is a common situation when the equation is semilinear and the a priori estimate or assumption on the solution has subcritical scaling.
+== Existence and uniqueness results ==
+For a variety of nonlinear elliptic and parabolic equations, the existence of [[viscosity solutions]] can be obtained using [[Perron's method]]. The uniqueness of solutions is a consequence of the [[comparison principle]].
-=== A slightly more complicated example ===
+There are some equations for which this general framework does not work, for example the [[surface quasi-geostrophic equation]]. One could say that the underlying reason is that the equation is not ''purely'' parabolic, but it has one hyperbolic term.
-Imagine now that we have a fractional conservation law of the form
+== Regularity results ==
-\[ u_t + (-\Delta)^s u + \mathrm{div} \ F(\nabla u) = 0. \]
-Where $F$ is some smooth vector valued function and $s \in (0,1/2)$. Assume that a solution $u$ is known to be $C^\alpha$ for some $\alpha>1-2s$. As before, $u$ coincides with the solution $v$ of a linear equation whose coefficients depend on $u$. However, the equation is now more complicated.
-\begin{align*}
-v(0,x)&=u(0,x) \\
- v_t + (-\Delta)^s v  + b(x,t) \cdot \nabla v &= 0
-\end{align*}
-where $b(x,t) = F'(u)$. Since $F$ is smooth and $u \in C^\alpha$ in space, we have that $b \in C^\alpha$ in space, which implies that $v \in C^{1,\alpha}$ in space by the estimates for linear [[drift-diffusion equations]]. Therefore $u \in C^{1,\alpha}$. Differentiating the equation and repeating the procedure we get $u \in C^{2,\alpha}$, $u \in C^{3,\alpha}$, etc...
-The procedure is slightly more complicated because the linear equation has variable coefficients and a less standard estimate for linear equations is used. Still the outline of the idea is the same. Bootstrap arguments are considered to be automatic once we have an priori estimate which is sufficient for a stronger regularity result for linear equations with coefficients.
+The regularity tools used for nonlocal equations vary depending on the type of equation.
+=== Nonlinear equations ===
+The starting point to study the regularity of solutions to a nonlinear elliptic or parabolic equation are the [[Holder estimates]] which hold under very weak assumptions and rough coefficients. They are related to the [[Harnack inequality]].
+For some [[fully nonlinear integro-differential equation]] with continuous coefficients, we can prove [[differentiability estimates|$C^{1,\alpha}$ estimates]].
+Under certain hypothesis, the nonlocal [[Bellman equation]] from optimal stochastic control has classical solutions due to the [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]].
+=== Semilinear equations ===
+There are several interesting models that are [[semilinear equations]]. Those equations consists of either the [[fractional Laplacian]] or [[fractional heat equation]] plus a nonlinear term.
+There are challenging regularity questions especially when the Laplacian interacts with gradient terms in [[Drift-diffusion equations]]. A simple method that has been successful in proving the well posedness of some semilinear equations with drift terms in the critical case (when both terms have the same scaling properties) is the [[conserved modulus of continuity approach]], often called "nonlocal maximum principle method".