Local well-posedness for hyperbolic systems of equations with fractional dissipation

Abstract

We establish a local well-posedness theory for a class of hyperbolic quasilinear evolution systems with fractional dissipation and commutators of the fractional Laplacian $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$. The analysis is motivated by the compressible isentropic Navier-Stokes equations with fractional diffusion and the compressible Euler alignment system with singular communication weights. Our approach does not rely on the cancellation condition previously used to recover the coercivity of the highest-order non-local terms. Instead, we exploit the self-adjointness of the fractional Laplacian together with an elementary algebraic identity for commutators of Fourier multipliers. Then, we use the non-homogeneous Littlewood-Paley decomposition to develop some new commutator estimates. These ingredients yield the coercive structure of the non-local terms, thus showing that the contribution of the commutators to the energy is absorbed by the dissipation. Our method remains applicable for systems with variable coefficients. We further show that the associated solution operator fails to satisfy Banach's contraction principle. To overcome this difficulty, we establish a H\"older continuity estimate of order $1/2$, combine it with the Aubin-Lions compactness theorem and a tail-control argument of the $L^{2}$ norm, that yields the relative compactness of the solution operator, allowing the application of Schauder's fixed point theorem. The resulting theory provides local existence for arbitrary orders $\alpha\in(0,1)$ and identifies the obstacles for the local uniqueness of solutions in the presence of fractional commutators.