Abstract
We introduce a notion of global weak solution to the Navier–Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces \({\dot{B}^{-1+\frac{3}{p}}_{p,\infty}}\), p > 3. These solutions satisfy a certain stability property with respect to the weak-\({\ast}\) convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.
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Acknowledgements
The first author was partially supported by the NDSEG Graduate Fellowship. He also thanks his advisor, Vladimír Šverák, as well as Simon Bortz and Raghavendra Venkatraman for helpful suggestions. The second author was supported by an EPSRC Doctoral Prize award. We thank P. G. Lemarié-Rieusset for pointing out the reference [35], as well as the anonymous referee for his or her work reviewing the paper.
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Albritton, D., Barker, T. Global Weak Besov Solutions of the Navier–Stokes Equations and Applications. Arch Rational Mech Anal 232, 197–263 (2019). https://doi.org/10.1007/s00205-018-1319-0
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DOI: https://doi.org/10.1007/s00205-018-1319-0