2023/07/21 by Myong‐Hwan Ri, Ri, Myong-Hwan · 1 citation
Mathematics · Physics and Astronomy · Engineering · #Navier-Stokes equation solutions #Cosmology and Gravitation Theories #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2307.11312
We show that a Leray-Hopf weak solution to the 3D Navier-Stokes Cauchy problem belonging to the space L^∞(0,T; B-1∞,∞(\mathbb R3)) is regular in (0,T]. As a consequence, it follows that any Leray-Hopf weak solution to the 3D Navier-Stokes equations is regular while it is temporally bounded in the largest critical space B-1∞,∞(\mathbb R3) as well as in any critical spaces. For the proof we present a new elementary method which is to superpose the energy norm of high frequency parts in an appropriate way to generate higher order norms. Thus, starting from the energy estimates of high frequency parts of a weak solution, one can obtain its estimates of higher order norms. By a linear energy superposition we get very simple and short proofs for known regularity criteria for Leray-Hopf weak solutions in endpoint Besov spaces Bσ∞,∞ for σ∈ [-1,0), the extension of Prodi-Serrin conditions. The main result of the paper is proved by applying technique of a nonlinear energy superposition and linear energy superpositions, repeatedly. The energy superposition method developed in the paper can also be applied to other supercritical nonlinear PDEs.