Flexibility throughout the subcritical integrability range

Extend flexibility for inviscid surface quasi-geostrophic momentum weak solutions to every exponent 4/3 <= p < 3, with active scalar in C([0,1];L^p(T^2)) and endpoint flexibility in L^p(T^2), thereby constructing Hamiltonian-nonconserving weak solutions with integrability arbitrarily close to L^3.

Background

The main theorem establishes endpoint flexibility only for the explicit exponent p-bar = 4/3 + 10^-5. The paper notes that L^3 is a natural upper threshold because space-time L^3 integrability guarantees Hamiltonian conservation, while the constructed solutions can fail to conserve the Hamiltonian.

The authors explicitly leave open whether the flexibility mechanism can cover the entire subcritical interval from the concentration-critical exponent 4/3 up to, but excluding, 3. Such an extension would substantially enlarge the known range of Hamiltonian-nonconserving weak solutions.