Work

Upper bounds on homogeneous fractional Gagliardo–Nirenberg–Sobolev constants via parabolic estimates

Year: 2025

Type: article

Abstract: Common proofs of the Gagliardo–Nirenberg–Sobolev (GNS) inequalities do not provide explicit bounds on the involved constants, unless a sharp constant is being determined. GNS inequalities naturally occur in error estimates for numerical approximations. In particular, bounds on GNS constants allow us to provide explicit a priori estimates. We provide an algorithm that determines explicit upper bounds on the non-endpoint homogeneous GNS inequalities in terms of explicit upper bounds for Young’s convolution inequality and parabolic estimates. Our method is based on the heat-kernel representation of the inverse Laplacian, from which we deduce interpolation estimates.

Source: International Journal of Mathematics

DOI: 10.1142/s0129167x26500163

Author: Michael Hott

Institution: Mathematical Sciences Research Institute

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, University of Texas at Austin, Division of Mathematical Sciences

Awards: Mathematical Analysis of Dispersion and Transport in Quantum Dynamics, G3324802687, G4713160749, Mathematical Analysis of the Dynamics of Complex Quantum Systems, G8223560616