Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

Camilli, Fabio; Goffi, Alessandro

doi:10.1007/s00030-020-0624-0

Cited by 14 publications

(16 citation statements)

References 103 publications

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“…An important consequence of the variational interpretation of the Mean Field Games is the possibility to show the existence of a solution to system (1.1) under weak regularity assumptions. With smoothing coupling term more regular solutions may be constructed using results from recent work on time-fractional Hamilton-Jacobi equations [33]. One interesting direction will be to study the long time behavior of such time-fractional MFG systems.…”

Section: Discussionmentioning

confidence: 99%

“…The limit (m, w) is then the minimizer of B(m, w). Existence and uniqueness of a classical solution to (3.5) (and therefore to (3.4)) has been recently studied in [33]. If there exists a classical solution to the backward HJ equation in (1.1), then the vector field ∇u is regular and also the FP equation admits a unique classical solution.…”

Section: Time-fractional Mean Field Gamesmentioning

confidence: 99%

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Variational Time-Fractional Mean Field Games

Tang

Camilli

2019

Dyn Games Appl

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We consider the variational structure of a time-fractional second order Mean Field Games (MFG) system. The MFG system consists of time-fractional Fokker-Planck and Hamilton-Jacobi-Bellman equations. In such a situation the individual agent follows a non-Markovian dynamics given by a subdiffusion process. Hence, the results of this paper extend the theory of variational MFG to the subdiffusive situation.

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Section: Discussionmentioning

confidence: 99%

Section: Time-fractional Mean Field Gamesmentioning

confidence: 99%

Variational Time-Fractional Mean Field Games

Tang

Camilli

2019

Dyn Games Appl

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“…In [11], Giga and Namba investigated the well-posedness of Hamilton-Jacobi equations with a Caputo fractional time derivative, with a main purpose of finding a proper notion of viscosity solutions so that the underlying Hamilton-Jacobi equation is well-posed. A further study along this line is recently provided by Camilli and Goffi [12]. Their study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.…”

Section: Introductionmentioning

confidence: 99%

How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

Quan¹

2020

CSIAM-AM

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There exists a well defined energy for classical phase-field equations under which the dissipation law is satisfied, i.e., the energy is non-increasing with respect to time. However, it is not clear how to extend the energy definition to time-fractional phase-field equations so that the corresponding dissipation law is still satisfied. In this work, we will try to settle this problem for phase-field equations with Caputo timefractional derivative, by defining a nonlocal energy as an averaging of the classical energy with a time-dependent weight function. As the governing equation exhibits both nonlocal and nonlinear behavior, the dissipation analysis is challenging. To deal with this, we propose a new theorem on judging the positive definiteness of a symmetric function, that is derived from a special Cholesky decomposition. Then, the nonlocal energy is proved to be dissipative under a simple restriction of the weight function. Within the same framework, the time fractional derivative of classical energy for timefractional phase-field models can be proved to be always nonpositive.

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“…The theory of viscosity solution for this kind of equations has been constructed by Topp and Yangari [30], Giga, Namba et al [14,25]. Recently, Camilli and Goffi studied the existence and uniqueness of classical solutions to the time-fractional Hamilton-Jacobi-Bellman equation [9]. The time-fractional Fokker-Planck equation can be more suitably considered in the variational formulation, and the weak solution is in the sense of distributions.…”

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confidence: 99%

“…for a given vector U = (U n i,j ) 1≤n≤N . Collecting (9) and 11, we end up with the following scheme for (1)…”

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confidence: 99%

Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

Camilli¹,

Duisembay²,

Tang³

2021

JDG

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In this paper, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem.

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Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

Cited by 14 publications

References 103 publications

Variational Time-Fractional Mean Field Games

Variational Time-Fractional Mean Field Games

How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

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