# Introduction to nonlocal equations

(Difference between revisions)
 Revision as of 04:08, 7 February 2012 (view source)Nestor (Talk | contribs) (Created page with " == Existence and uniqueness results == For a variety of nonlinear elliptic and parabolic equations, the existence of viscosity solutions can be obtained using [[Perron's met...")← Older edit Latest revision as of 18:34, 5 May 2014 (view source)Xavi (Talk | contribs) (6 intermediate revisions not shown) Line 1: Line 1: + Traditional partial differential equations are relations between the values of an unknown function and its derivatives of different orders. In order to check whether a partial differential equation holds at a particular point, one needs to known only the values of the function in an arbitrarily small neighborhood, so that all derivatives can be computed. A nonlocal equation is a relation for which the opposite happens. In order to check whether a nonlocal equations holds at a point, information about the values of the function far from that point is needed. Most of the times, this is because the equation involves integral operators. A simple example could be + $u_t(t,x) = \int_{\R^d} (u(t,x+y)-u(t,x)) k(y) dy$ + for some kernel $k$. - == Existence and uniqueness results == + When dealing with nonlocal equation, one can still talk about elliptic, parabolic and hyperbolic ones. This classification is vague, as much as it is for usual partial differential equations. An equation is elliptic when it fulfills some characteristics that are common to elliptic equations. Some properties of elliptic equations, for example, are the [[maximum principle]] and the interior [[regularity results for nonlocal equations|regularity results]]. These equations, being local or nonlocal, describe a function which does not evolve with time. The equation describes a situation in which the values of the function at every point equal some form of weighted average of other values of the function. There are several settings in which that kind of equations occur. An important example is the problems in [[stochastic control]], which motivate the study of [[fully nonlinear integro-differential equations]]. The generator of a [[Levy processes]] is a [[linear integro-differential operator]]. The Laplace equation is the prime example of a classical elliptic equation. Likewise, the equations involving the [[fractional Laplacian]] are the prime example of nonlocal elliptic equations. - 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]]. + $(-\Delta)^s u = f.$ - 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. + The parabolic equations are those which describe the evolution of a function which tries to converge to its equilibrium because of the effect of dissipation (in a broad sense). Much of the theory developed for elliptic equations can be extended to parabolic equations, but one often encounters nontrivial difficulties in the process. In these equations, we also find some form of [[maximum principle]] and [[regularity results for nonlocal equations|regularization results]]. - == Regularity results == + Most of this wiki is devoted to elliptic and parabolic equations: their solvability and regularity issues. - The regularity tools used for nonlocal equations vary depending on the type of equation. + To a great extent, the study of nonlocal equations is motivated by real world [[applications]]. - === Nonlinear equations === + The nonlocality in the equation can have different sources. The most common is perhaps to study nonlocal diffusions, often given by a term in the equation which is an [[linear integro-differential operator]]. This diffusion may interact with other terms, as in the [[drift-diffusion 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]]. + $u_t + b \cdot \nabla u + (-\Delta)^s u = 0.$ - For some [[fully nonlinear integro-differential equation]] with continuous coefficients, we can prove [[differentiability estimates|$C^{1,\alpha}$ estimates]]. + Another general family of nonlocal equations are the [[active scalar equations]]. These are equations which determine the evolution of a scalar quantity and have a drift which depends on the value of the solution via an integral operator. + \begin{align*} + \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \\ + u = \int \theta(y) K(x-y) dy + \end{align*} - 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]]. + Some equations from fluid mechanics are nonlocal. [[Boltzmann equation]] is a clear example. Strictly speaking, even the classical Navier-Stokes equation (governing the evolution of velocity in a viscous fluid) is a nonlocal equation due to the presence of the pressure. The gradient of pressure term effectively acts as a nonlocal operator that projects the drift term into the space of divergence free vector fields. + $u_t + \mathbb P [u \cdot \nabla u] - \Delta u = 0.$ - === Semilinear equations === + '''Further reads''' - 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. + * [[Starting page]] + * [[List of equations]] + * [[Open problems]] + * [[Semilinear equations]] + * [[Levy processes]] and [[stochastic control]] + * [[Fully nonlinear integro-differential equations]] - 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". + + {{stub}}

## Latest revision as of 18:34, 5 May 2014

Traditional partial differential equations are relations between the values of an unknown function and its derivatives of different orders. In order to check whether a partial differential equation holds at a particular point, one needs to known only the values of the function in an arbitrarily small neighborhood, so that all derivatives can be computed. A nonlocal equation is a relation for which the opposite happens. In order to check whether a nonlocal equations holds at a point, information about the values of the function far from that point is needed. Most of the times, this is because the equation involves integral operators. A simple example could be $u_t(t,x) = \int_{\R^d} (u(t,x+y)-u(t,x)) k(y) dy$ for some kernel $k$.

When dealing with nonlocal equation, one can still talk about elliptic, parabolic and hyperbolic ones. This classification is vague, as much as it is for usual partial differential equations. An equation is elliptic when it fulfills some characteristics that are common to elliptic equations. Some properties of elliptic equations, for example, are the maximum principle and the interior regularity results. These equations, being local or nonlocal, describe a function which does not evolve with time. The equation describes a situation in which the values of the function at every point equal some form of weighted average of other values of the function. There are several settings in which that kind of equations occur. An important example is the problems in stochastic control, which motivate the study of fully nonlinear integro-differential equations. The generator of a Levy processes is a linear integro-differential operator. The Laplace equation is the prime example of a classical elliptic equation. Likewise, the equations involving the fractional Laplacian are the prime example of nonlocal elliptic equations. $(-\Delta)^s u = f.$

The parabolic equations are those which describe the evolution of a function which tries to converge to its equilibrium because of the effect of dissipation (in a broad sense). Much of the theory developed for elliptic equations can be extended to parabolic equations, but one often encounters nontrivial difficulties in the process. In these equations, we also find some form of maximum principle and regularization results.

Most of this wiki is devoted to elliptic and parabolic equations: their solvability and regularity issues.

To a great extent, the study of nonlocal equations is motivated by real world applications.

The nonlocality in the equation can have different sources. The most common is perhaps to study nonlocal diffusions, often given by a term in the equation which is an linear integro-differential operator. This diffusion may interact with other terms, as in the drift-diffusion equations. $u_t + b \cdot \nabla u + (-\Delta)^s u = 0.$

Another general family of nonlocal equations are the active scalar equations. These are equations which determine the evolution of a scalar quantity and have a drift which depends on the value of the solution via an integral operator. \begin{align*} \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \\ u = \int \theta(y) K(x-y) dy \end{align*}

Some equations from fluid mechanics are nonlocal. Boltzmann equation is a clear example. Strictly speaking, even the classical Navier-Stokes equation (governing the evolution of velocity in a viscous fluid) is a nonlocal equation due to the presence of the pressure. The gradient of pressure term effectively acts as a nonlocal operator that projects the drift term into the space of divergence free vector fields. $u_t + \mathbb P [u \cdot \nabla u] - \Delta u = 0.$