Year: 2026
Type: preprint
Abstract: We study the local well-posedness of the Cauchy problem for a class of hyperbolic quasilinear evolution systems with fractional dissipation and commutators of the fractional Laplacian $(-Δ)^α$ with $α\in(0,1)$. The analysis is motivated by the compressible isentropic Navier-Stokes equations with fractional diffusion and the compressible Euler alignment system with singular communication weights. We show that the contribution of the commutators to the energy is absorbed by the dissipation. Our approach does not rely on the cancellation condition previously used to recover the coercivity of the highest-order non-local terms. We further show that the commutator term induces a loss of regularity in the stability estimates, preventing the standard contraction argument from being applied. Therefore, we instead establish the relative compactness of the solution operator and apply Schauder's fixed point theorem. The resulting theory provides local existence for arbitrary orders $α\in(0,1)$ and identifies the obstacles to local uniqueness in the presence of fractional commutators.
Source: arXiv (Cornell University)
Author: Angeles, Felipe
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166190374