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

  title={Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative},
  author={Fabio Camilli and Alessandro Goffi},
  journal={Nonlinear Differential Equations and Applications NoDEA},
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. Our 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. 
Existence, symmetries, and asymptotic properties of global solutions for a fractional diffusion equation with gradient nonlinearity
This work gives sufficient conditions to obtain the existence, positivity, symmetry, asymptotic and spatial behaviors of global solutions of a fractional reaction–diffusion equation with power-typeExpand
On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation
We consider an initial-boundary value problem for the time-fractional diffusion equation. We prove the equivalence of two notions of weak solutions, viscosity solutions and distributional solutions.
Approximation of an optimal control problem for the time-fractional Fokker-Planck equation
The scheme for the Fokker-Planck equation is constructed such that the duality structure of the PDE system is preserved on the discrete level and it is proved the well posedness of the scheme and the convergence to the solution of the continuous problem. Expand
Variational Time-Fractional Mean Field Games
The theory of variational MFG is extended to the subdiffusive situation in which the individual agent follows a non-Markovian dynamics given by a subdiffusion process. Expand
The well-posedness and exact solution of fractional magnetohydrodynamic equations
For numerous fluids between elastic and viscous materials, the fractional magnetohydrodynamic models have an advantage over the integer-order models. We study the fractional magnetohydrodynamicExpand
A P ] 2 J ul 2 01 9 Variational time-fractional Mean Field Games
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-JacobiBellman equations.Expand
How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations
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 clearExpand
Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations
We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As forExpand
Numerical Energy Dissipation for Time-Fractional Phase-Field Equations
This article studies in this article the energy dissipation of some numerical schemes for time-fractional phase-field models, including the convex-splitting scheme, the stabilization scheme, and the scalar auxiliary variable scheme. Expand
Inverse initial problem for fractional reaction-diffusion equation with nonlinearities
The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at theExpand


Lipschitz regularity for viscous Hamilton-Jacobi equations with L terms
We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, andExpand
Weakly coupled systems of parabolic Hamilton–Jacobi equations with Caputo time derivative
In this paper we prove existence and uniqueness of bounded viscosity solutions of weakly coupled systems of parabolic Hamilton–Jacobi equations with nonlocal ingredients, where the time evolution ofExpand
A Parabolic Problem with a Fractional Time Derivative
We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi–Nash–Moser Hölder regularity theorem for solutions in aExpand
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 aExpand
Adjoint and Compensated Compactness Methods for Hamilton–Jacobi PDE
We investigate the vanishing viscosity limit for Hamilton–Jacobi PDE with nonconvex Hamiltonians, and present a new method to augment the standard viscosity solution approach. The main idea is toExpand
Existence and uniqueness for parabolic problems with Caputo time derivative
Abstract In this paper we are interested in the well-posedness of fully nonlinear Cauchy problems in which the time derivative is of Caputo type. We address this question in the framework ofExpand
On an optimal control problem of time-fractional advection-diffusion equation
We consider an optimal control problem of an advection-diffusion equation with Caputo time-fractional derivative. By convex duality method we obtain as optimality condition a forward-backward coupledExpand
On Maximal Regularity for Abstract Parabolic Problems with Fractional Time Derivative
  • D. Guidetti
  • Mathematics
  • Mediterranean Journal of Mathematics
  • 2019
We consider initial value problems for abstract evolution equations with fractional time derivative. Concerning the Caputo derivative $$\mathbb {D}^\alpha u$$Dαu, we show that certain assumptions,Expand
Mild solutions to the time fractional Navier-Stokes equations in R-N
Abstract This paper addresses the existence and uniqueness of mild solutions to the Navier–Stokes equations with time fractional differential operator of order α ∈ ( 0 , 1 ) . Several interestingExpand
Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations
The purpose is to study the Cauchy problem for non-linear in time and space pseudo- differential equations. These include the fractional in time versions of Hamilton-Jacobi-Bellman (HJB) equationsExpand