Local \(L^2\) theory of the fractional Navier-Stokes equations and the self-similar solution. (English) Zbl 1509.35185
Summary: This paper is concerned with two aspects of the fractional Navier-Stokes equation. First, we establish the local \(L^2\) theory of the hypo-dissipative Navier-Stokes system. More precisely, the existence of local in time as well as global in time local energy weak solutions to the hypo-dissipative Navier-Stokes system is proved. In particular, in order to construct a pressure with an explicit representation, some technical innovations are required due to the lack of known results on the local regularity of the non-local Stokes operator. Secondly, as an important application to the local \(L^2\) theory, we give a second construction of large self-similar solutions of the hypo-dissipative Navier-Stokes system along with the Leray-Schauder degree theory.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
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