Forward self-similar solutions of the Navier–Stokes equations in the half space

Abstract

For the incompressible Navier-Stokes equations in the 3D half space, we show the existence of forward self-similar solutions for arbitrarily large self-similar initial data.

Discretely self-similar solutions to the Navier–Stokes equations with data in Lloc2 satisfying the local energy inequality

“…In [11], Jia andŠverák constructed forward selfsimilar solutions for large data where the data is assumed to be Hölder continuous away from the origin. This result has been generalized in a number of directions by a variety of authors [3,4,5,8,16,18,23]. This paper can be understood in the context of [3,8,18] and we briefly recall the main results of these papers.…”

Forward self-similar solutions of the fractional Navier-Stokes equations

“…When α = 1, Jia-Sverak prove the existence of forward self-similar solution and the associate wise-point estimates (1.8) in [17]. Korobkov-Tsai [23] give another proof the existence of forward self-similar solution was shown in [23] via the blowup argument. But they did not show that the solution has the decay estimate.…”

“…In contrast with the case of backward self-similar solutions, several results of nontrivial forward self-similar solutions were established in the past years. In [7,8], Cannone-Meyer-Planchon firstly proved the existence and uniqueness of the small forward self-similar solutions in the framework of homogeneous Besov spaces, see also for examples Barraza [3] in Lorentz space L (3,∞) (R 3 ), and Koch and Tataru [23] in BMO −1 (R 3 ).…”

Short Time Regularity of Navier–Stokes Flows with Locally L3 Initial Data and Applications

“…In [15], Jia andŠverák constructed forward self-similar solutions for large data where the data is assumed to be Hölder continuous away from the origin. This result has been generalized in a number of directions by a variety of authors [2,3,4,5,7,21,23,32]; see also the survey [17].…”