A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

On Right Continuity in $(L^2)^3$ at All the Points of Energy-regularized Solutions for the 3D Navier-Stokes Equations

“…where z := Π s,T z ∈ C([s, T ]; H w ) ∩ L 2 (s, T ; V ), and dz dt is the derivative of z in the sense of distributions D * ((s, T ); V * ). When s = τ such problem was considered in [17,11,12,14,15,18,26] and references…”

Leray-Hopf and Continuity Properties for All Weak Solutions for the 3D~Navier-Stokes Equations

“…in the sense of distributions; cf. Kapustyan et al [9,10]; Kasyanov et al [11,12]; Melnik and Toscano [14]; Zgurovsky et al [19,Chapter 6].…”

Regularity Criteria For Strong Solutions To The 3D Navier-Stokes Equations

“…Two of the profound open problems in the theory of three dimensional viscous flows are the unique solvability theorem for all time and the regularity of solutions. For the three-dimensional Navier-Stokes system weak solutions are known to exist by a basic result by Leray from 1934 [10], but the uniqueness is still open problem [1]- [3] and [8]. Furthermore, the strong solutions for the 3D Navier-Stokes equations are unique and can be shown to exist on a certain finite time interval for small initial data and small forcing term, but the global regularity for the 3D Navier-Stokes is still open problems (see [4]- [8], [12]- [14] and references therein).…”

“…For the three-dimensional Navier-Stokes system weak solutions are known to exist by a basic result by Leray from 1934 [10], but the uniqueness is still open problem [1]- [3] and [8]. Furthermore, the strong solutions for the 3D Navier-Stokes equations are unique and can be shown to exist on a certain finite time interval for small initial data and small forcing term, but the global regularity for the 3D Navier-Stokes is still open problems (see [4]- [8], [12]- [14] and references therein). In 1933 [9], Leray showed that in the absence of forcing (f = 0), all solutions of Navier-Stokes equations are eventually smooth (i.e.…”