Igor Kukavica

One component regularity for the Navier–Stokes equations

We establish sufficient conditions for the regularity of solutions of the Navier–Stokes system based on conditions on one component of the velocity. The first result states that if , where and 54/23 ≤ r ≤ 18/5, then the solution is regular. The second result is that if , where and 24/5 ≤ r ≤ ∞, then the solution is regular. These statements improve earlier results on one component regularity.

Navier-Stokes equations with regularity in one direction

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirao da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.

On the Regularity of the Primitive Equations of the Ocean

We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography. Previously, the existence of global strong solutions was known in the case of the Neumann boundary conditions in a cylindrical domain (Cao and Titi 2007 Ann. Math. 166 245–67).

On the radius of analyticity of solutions to the three-dimensional Euler equations

We address the problem of analyticity of smooth solutions u of the incompressible Euler equations. If the initial datum is real-analytic, the solution remains real-analytic as long as f t 0 ∥∇u(·, s)∥ L∞ ds < ∞. Using a Gevrey-class approach we obtain lower bounds on the radius of space analyticity which depend algebraically on exp ∫ t 0 ∥∇u(·,s)∥ L∞ ds. In particular, we answer in the positive a question posed by Levermore and Oliver.

On the inviscid limit of the Navier-Stokes equations

We consider the convergence in the L 2 norm, uniformly in time, of the Navier-Stokes equations with Dirichlet boundary conditions to the Euler equations with slip boundary conditions. We prove that if the Oleinik conditions of no back-flow in the trace of the Euler flow, and of a lower bound for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then the inviscid limit holds.

ON THE LOCAL WELL-POSEDNESS OF THE PRANDTL AND HYDROSTATIC EULER EQUATIONS WITH MULTIPLE MONOTONICITY REGIONS ∗

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, if the initial datum

Determining nodes for the Kuramoto-Sivashinsky equation

u_0

On the number of determining nodes for the Ginzburg-Landau equation

is monotone on a number of intervals (on some strictly increasing, on some strictly decreasing) and analytic on the complement of these intervals, we show that the local existence and uniqueness hold. The same result is true for the hydrostatic Euler equations if we assume these conditions for the initial vorticity

On well-posedness for a free boundary fluid-structure model

\omega_0=\partial_y u_0

On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations

.