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In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 (1) 1 (R n) , which corresponds to a critical case of the generalized Campanato spaces L q (N) s (R n). The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 (R n) ↪ L 1 (1) 1 (R n) ↪ C 0 , 1 (R n) , where B ∞ , 1 1 (R n) and C 0 , 1 (R n) are the Besov space and the Lipschitz space respectively. In particular L 1 (1) 1 (R n) contains non- C 1 (R n) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 (1) 1 (R n) , for which the solution to the Euler equations blows up in finite time. [ABSTRACT FROM AUTHOR]

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the L 2 norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale. [ABSTRACT FROM AUTHOR]

In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies ∫ 0 t ∗ (t ∗ - t) ‖ ω (t) ‖ L ∞ (B (x ∗ , R 0)) d t < + ∞ for a ball B (x ∗ , R 0) away from the axis of symmetry, then there exists no singularity at t = t ∗ in the torus T (x ∗ , R) generated by rotation of the ball B (x ∗ , R 0) around the axis. This implies that possible singularity at t = t ∗ in the torus T (x ∗ , R) is excluded if the vorticity satisfies the blow-up rate ‖ ω (t) ‖ L ∞ (T (x ∗ , R)) = O 1 (t ∗ - t) γ as t → t ∗ , where γ < 2 , and the torus T (x ∗ , R) does not touch the axis. [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]