2022/11/23 by Diego Chamorro, Chamorro, Diego, Bruno Poggi +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2211.13077
openalex publication_date 2022/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (-Δ)^\fracα2 with 0<α<2. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space H^\fracα2(ℝ) 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.