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

Diego Chamorro (1) , Bruno Poggi (2)

1 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
2 UAB - Departament de Matemàtiques [Barcelona]

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 Ḣ α 2 (R 3) 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, however the study of regularity in the regime 0 < α ≤ 1 seems for the moment to be an open problem.

Mots clés: Liouville type theorems; Fractional Navier-Stokes equations. Mathematics Subject Classification: 76D03; 35A02; 35B65

Domaines: Equations aux dérivées partielles [math.AP]

Diego Chamorro, Bruno Poggi. On an almost sharp Liouville type theorem for fractional Navier-Stokes equations. 2022. ⟨hal-03867798⟩