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We consider the Navier–Stokes equations in R 3 subject to the initial condition with initial velocity field in L loc 2 (R 3) such that lim sup R → + ∞ R − 1 ‖ u 0 ‖ L 2 (B (R)) < + ∞. Our aim is to show the local existence of a weak solution, global existence of weak solution if C = 0 and the partial regularity in the sense of Caffarelli–Kohn–Nirenberg. [ABSTRACT FROM AUTHOR]

We prove the existence of a forward discretely self-similar solutions to the Navier–Stokes equations in R 3 × ( 0 , + ∞ ) for a discretely self-similar initial velocity belonging to L l o c 2 ( R 3 ) . [ABSTRACT FROM AUTHOR]

We study the scenario of discretely self-similar blowup for Navier–Stokes equations. We prove that at the possible blow-up time such solutions have only one point singularity. In case of the scaling parameter λ near 1, we remove the singularity. [ABSTRACT FROM PUBLISHER]

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ru̇žička-Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $${L^{p, \infty} (\mathbb{R}^3)}$$ with $${p > \frac{3}{2}}$$ . [ABSTRACT FROM AUTHOR]

In this paper we prove three different Liouville type theorems for the steady Navier–Stokes equations in R 3 . In the first theorem we improve logarithmically the well-known L 9 2 ( R 3 ) result. In the second theorem we present a sufficient condition for the trivially of the solution ( v = 0 ) in terms of the head pressure, Q = 1 2 | v | 2 + p . The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee v = 0 . [ABSTRACT FROM AUTHOR]

The present paper deals with the problem of local regularity of weak solutions to the Navier–Stokes equation in Ω × ( 0 , T ) with forcing term f in L 2 . We prove that u is strong in a sub-cylinder Q r ⊂ Ω × ( 0 , T ) if two velocity components u 1 , u 2 satisfy a Serrin-type condition. [ABSTRACT FROM AUTHOR]

In the present paper we prove that a weak Leray solution to the Navier-Stokes equations in R³ × (0, T] is regular provided the gradient of one component belongs to the Prodi-Serrin class ∇u³ Є L4(0, T; L²). [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary Navier–Stokes equations in R 3 . Let V = (V ij) be a potential function of a smooth solution u, which means u = ∇ · V . We show that if there exists 3 < s < + ∞ such that the L s mean oscillation of the potential function has certain growth condition near infinity, namely 1 | B (r) | ∫ B (r) | V - V B (r) | s d x ≤ C r min { s - 3 3 , s 6 } ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]