Introduction to nonlocal equations

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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 \[ u_t(t,x) = \int_{\R^d} (u(t,y)-u(t,x)) k(y) dy \] 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. \[ u_t + \mathbb P [u \cdot \nabla u] - \Delta u = 0.\]

Further reads

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