The local Cauchy problem for hyperbolic systems of equations with fractional dissipation
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∈(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∈(0,1) and identifies the obstacles to local uniqueness in the presence of fractional commutators.
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BibTeX
@misc{angeles2026local,
title = {{The local Cauchy problem for hyperbolic systems of equations with fractional dissipation}},
author = {Felipe Angeles},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.26887},
eprint = {2606.26887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26887},
}
RIS
TY - UNPB
TI - The local Cauchy problem for hyperbolic systems of equations with fractional dissipation
AU - Felipe Angeles
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2606.26887
UR - https://arxiv.org/abs/2606.26887
AB - 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∈(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∈(0,1) and identifies the obstacles to local uniqueness in the presence of fractional commutators.
ER -