Research on time fractional gradient flows
Time fractional gradient flows. Joint work with Abner J. Salgado (UTK, USA).
Link to our paper: Time fractional gradient flows: Theory and numerics.
Our work Time fractional gradient flows: Theory and numerics extends the a posteriori error estimates in (R. H. Nochetto, G. Savaré, C. Verdi 2000 Comm. Pure Appl. Math.) for the classical gradient flow problems to the fractional time setting. We condier the problem
\[D_c^\alpha u(t) + \mathcal{A}(u(t)) \ni f(t), t\in (0,T], \quad u(0) = u_0,\]where $u: [0,T] \to \mathcal{H}$ for a separable Hilbert space $\mathcal{H}$ and $\mathcal{A} = \partial \Phi$ is the subdifferential of a convex functional $\Phi$, which is bounded from below. Here $D_c^\alpha$ stands for the Caputo fractional time derivative of order $\alpha \in (0,1)$, which is defined as
\[D_c^\alpha u(t) := \frac1{\Gamma(1-\alpha)} \int_0^t \frac{u'(r)}{(t-r)^\alpha} dr\]for differentiable function $u$. This definition could also be generalized to a wider class of $u$.
Among our contributions, I want to highlight two results specifically. One is the fact that the hat functions associated with the fractional time derivative are non-negative. The proof of this fact is based on careful analysis of the discrete fractional time derivative and an elegant trick. Another is the a priori convergence rate for $f \in L^2_{\alpha}$ without additional regularity on the derivative of $f$. Due to the nonlocality, the argument in (R. H. Nochetto, G. Savaré, C. Verdi 2000 Comm. Pure Appl. Math.) based on orthogonality could not be applied here.
In addition, we also prove the existence and uniqueness of the solutions under slightly weaker conditions compared to (L. Li, J.-G. Liu 2019 SIAM J. Numer. Anal.). Our assumptions on the data are different to those in (G. Akagi 2019 Israel J. Math.), where time fractional gradient flows are also studied. All of our existence, uniqueness, a priori, and a posteriori approximation results are also extended to the case in which we allow a Lipschitz perturbation of the subdifferential operator $\partial \Phi$.