The variational approach to s-fractional heat flows and the limit cases s-+0+and s-+1-

This paper deals with the limit cases for s-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as s-+ 0+ and s-+ 1-. We describe the fractional heat flows as minimizing move-ments of the corresponding Gagliardo seminorms, with re-spect to the L2 metric. To this end, we first provide a Gamma-convergence analysis for the s-Gagliardo seminorms as s-+ 0+ and s-+ 1-; then, we exploit an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of Gamma-converging uniformly lambda-convex energy function-als. We prove that s-fractional heat flows (suitably scaled in time) converge to the standard heat flow as s-+ 1-, and to a de-generate ODE type flow as s-+ 0+. Moreover, looking at the next order term in the asymptotic expansion of the s -fractional Gagliardo seminorm, we show that suitably forced s-fractional heat flows converge, as s-+ 0+, to the parabolic flow of an energy functional that can be seen as a sort of renormalized 0-Gagliardo seminorm: the resulting parabolic equation involves the first variation of such an energy, that can be understood as a zero (or logarithmic) Laplacian.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper studies the limit cases of s-fractional heat flows in a cylindrical domain with homogeneous Dirichlet boundary conditions, as s-+ 0+ and s-+ 1-. The fractional heat flows are described as minimizing movements of the corresponding Gagliardo seminorms with respect to the L2 metric. The results show that the s-fractional heat flows converge to the standard heat flow as s-+ 1- and to a degenerate ODE type flow as s-+ 0+. Additionally, the next order term in the asymptotic expansion of the s-fractional Gagliardo seminorm reveals that suitably forced s-fractional heat flows converge to a parabolic flow of an energy functional that can be seen as a sort of renormalized 0-Gagliardo seminorm, involving the first variation of such an energy.(c) 2023 Elsevier Inc. All rights reserved.