A microlocal lift of Navier-Stokes dynamics on manifolds yields an if-and-only-if geometric criterion for solution blow-up in terms of deformation integrability, directional entropy, and lifted energy.
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The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n=3..9 and supercritical q in [1, n/2).
Computational searches maximizing LPS integrals and L3 norms in 3D periodic Navier-Stokes flows found no evidence of singularity formation, but quantified close approaches and transient growth.
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On Geometric Evolution and Microlocal Regularity of the Navier-Stokes Equations
A microlocal lift of Navier-Stokes dynamics on manifolds yields an if-and-only-if geometric criterion for solution blow-up in terms of deformation integrability, directional entropy, and lifted energy.
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Spectral instability and non-uniqueness of mild solutions for the Keller-Segel system
The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n=3..9 and supercritical q in [1, n/2).
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The Ladyzhenskaya-Prodi-Serrin Conditions and the Search for Extreme Behavior in 3D Navier-Stokes Flows
Computational searches maximizing LPS integrals and L3 norms in 3D periodic Navier-Stokes flows found no evidence of singularity formation, but quantified close approaches and transient growth.