Heat kernels for a class of hybrid evolution equations

2020/07/21 by Nicola Garofalo, Garofalo, Nicola, Giulio Tralli +1 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2007.10862

Abstract

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form \mathscr L₁ + \mathscr L₂ - ∂_t, but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators \mathscr L₁ - ∂_t and \mathscr L₂ - ∂_t. Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's 1967 groundbreaking paper on hypoellipticity.

Cited by

• Nonexistence of global solutions to the Grushin heat equation with nonlocal and local nonlinearities
• Dispersive equations with invariant measures
• Local and global solvability of the Grushin heat equation with mixed nonlinear memory and reaction terms

Related

• Curvature terms in small time heat kernel expansion for a model class of hypoelliptic Hörmander operators
• A Hamiltonian Approach to the Heat Kernel of a SubLaplacian on S(2n+1)
• Evolution equations involving nonlinear truncated Laplacian operators
• Heat Kernel Analysis for Ornstein-Uhlenbeck Operators with Quadratic Potentials
• Kolmogorov-Fokker-Planck operators in dimension two: heat kernel and curvature