Fractional higher differentiability of solutions to strongly nonlinear Stokes systems

TL;DR

This paper investigates fractional regularity of solutions to nonlinear Stokes systems with non-polynomial operators, establishing membership in Besov and Orlicz-Besov spaces for the symmetric gradient and pressure.

Contribution

It introduces new fractional regularity results for solutions to nonlinear Stokes systems using Orlicz and Orlicz-Sobolev spaces, extending classical results to non-power-type nonlinearities.

Findings

Symmetric gradient belongs to Besov spaces under certain conditions.

Fractional regularity of pressure is characterized in Orlicz-Besov spaces.

Results apply to associated elliptic systems with similar regularity properties.

Abstract

This work concerns stationary Stokes type systems governed by a general class of non-necessarily power-type nonlinearities. Fractional regularity properties of the symmetric gradient of local solutions are established, depending on a balance between the nonlinearity of the differential operator and the degree of integrability of the datum on right-hand side. The non-polynomial character of the differential operators calls for the use of Orlicz and Orlicz-Sobolev spaces as an appropriate functional framework for both the solutions and the datum. The regularity result amounts to the membership of a nonlinear expression of the symmetric gradient in Besov spaces. Fractional regularity of the pressure term is also exhibited and is formulated in terms of Orlicz-Besov spaces. Fractional Sobolev regularity of the symmetric gradient and of the pressure follow as a consequence.} Parallel results for the symmetric gradient of local solutions to the associated plain elliptic system are also offered. A new version of a Poincar\'e-Sobolev inequality in Orlicz spaces, in modular form, on domains with finite measure plays a role in the proofs.