On an almost sharp Liouville-type theorem for fractional Navier-Stokes equations

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Diego Chamorro
Universit´e Paris-Saclay. Laboratoire de Math´ematiques et Mod´elisation d’Evry
https://orcid.org/0000-0002-3298-023X
Bruno Poggi
Universitat Autònoma de Barcelona. Departament de Matemàtiques
https://orcid.org/0000-0002-7992-8578

We investigate existence, Liouville-type theorems, and regularity results for the 3D stationary and incompressible fractional Navier–Stokes equations: in this setting the usual Laplacian is replaced by its fractional power (−∆) α 2 with 0 < α < 2. By applying a fixed-point argument, weak solutions can be obtained in the Sobolev space H˙α2 (R3) and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of α that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for 3/5 < α < 5/3. Moreover, in the case 1 < α < 2 a gain of regularity is established under some conditions, although the study of regularity in the regime 0 < α ≤ 1 seems for the moment to be an open problem.

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Liouville-type theorems, fractional Navier–Stokes equations

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Chamorro, Diego; Poggi, Bruno. «On an almost sharp Liouville-type theorem for fractional Navier-Stokes equations». Publicacions Matemàtiques, 2025, vol.VOL 69, núm. 1, p. 27-43, doi:10.5565/PUBLMAT6912502.
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(c) 2024

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