Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative

Abstract

A Hamilton-Jacobi equation with Caputo's time-fractional derivative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose comparison principle as well as Perron's method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time-fractional derivatives with variable coefficients.2010 Mathematics Subject Classification. 26A33; 35D40; 35F21.

A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time-fractional derivative

“…In this section, we briefly review definitions and some results for the continuous problem (1.1) (we refer to [8,13] for more details). For a function f :…”

“…Also, the numerical approximation of differential equations with fractional timederivative has been extensively analyzed [3,9,10]. Since in general smooth solutions to Hamilton-Jacobi equations are not expected to exist, for equation (1.1) a theory of weak solutions, in viscosity sense, has been introduced, in [8,13,18]. Most of the results and techniques which hold in the classical case, i.e., for α = 1, have been extended to the fractional case in order to prove the well-posedness of the Hamilton-Jacobi equation (1.1).…”

On a discrete scheme for time fractional fully nonlinear evolution equations

“…It has inspired further research on numerous related topics. We refer to a non-exhaustive list of references [10,14,2,3,7,15,1,13,9,4] and the references therein. Among these results, the authors of [2,1] mainly study regularity of solutions to a space-time nonlocal equation with Caputo's time fractional derivative in the framework of viscosity solutions.…”

“…More recently, unique existence of a viscosity solution to the initial value problem with Caputo's time fractional derivatives has been established in the thesis of Namba [12] and independently and concurrently by Topp and Yangari [15]. The main part of [12] on this subject has been published in [7,13]. For example, a comparison principle, Perron's method, and stability results for (1.1) in bounded domains with various boundary conditions have been established in [7,13].…”

“…We refer to [15, Lemma 2.3] and [13, Proposition 2.5] for proofs. Note that the original definition of viscosity solutions in[12,7] looks stronger but it turns out that it is the same [12, Lemma 2.9, Proposition 3.6].…”