  Provable nonlinear stability bounds the discrete approximation and ensures
that the discretization does not diverge. For high-order methods, discrete
nonlinear stability and entropy stability, have been successfully implemented
for discontinuous Galerkin (DG) and residual distribution schemes, where the
stability proofs depend on properties of L2-norms. In this paper, nonlinearly
stable flux reconstruction (NSFR) schemes are developed for three-dimensional
compressible flow in curvilinear coordinates. NSFR is derived by merging the
energy stable FR (ESFR) framework with entropy stable DG schemes. NSFR is
demonstrated to use larger time-steps than DG due to the ESFR correction
functions. NSFR differs from ESFR schemes in the literature since it
incorporates the FR correction functions on the volume terms through the use of
a modified mass matrix. We also prove that discrete kinetic energy stability
cannot be preserved to machine precision for quadrature rules where the surface
quadrature is not a subset of the volume quadrature. This paper also presents
the NSFR modified mass matrix in a weight-adjusted form. This form reduces the
computational cost in curvilinear coordinates through sum-fcatorization and
low-storage techniques. The nonlinear stability properties of the scheme are
verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green
vortex problem and the correct orders of convergence were obtained for a
manufactured solution. Lastly, we perform a computational cost comparison
between conservative DG, overintegrated DG, and our proposed entropy conserving
NSFR scheme, and find that our proposed entropy conserving NSFR scheme is
computationally competitive with the conservative DG scheme.
