Partly Dissipative Semigroups Generated by the Navier-Stokes System on Two-Dimensional Manifolds, and Their Attractors

Abstract

We examine recent claims for a considerable amount of leptogenesis, in some inflationary scenarios, through the gravitational anomaly in the lepton number current. We find that when the short distances contributions are properly included the amount of lepton number generated is actually much smaller.

Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations

“…Estimate (1.2) also holds for the Navier-Stokes system on a two-dimensional compact manifold (for example, S 2 ) and in a bounded domain with the so-called free boundary conditions [22]. We also note that, so far, no lower bounds are known for system (1.1) with Dirichlet boundary conditions.…”

“…Furthermore, we have orthogonality relations and formulas for the integration by parts [22] that are totally similar to the space-periodic case: Now, the existence of the semigroup S t : H → H and the global attractor A H is established as in the space-periodic case [22]. Taking the scalar product of (3.2) with Au and integrating by parts using (3.3) and (3.4), we obtain, as above, that the following inequality holds on the attractor A:…”

“…Next, the operator rot acts on tangent vectors u and scalars ψ (which are identified with normal vectors) as follows [22]:…”

“…For the sake of completeness, we give an invariant proof of (2.3) (in the sense that it also works for the sphere and a bounded domain with free boundary conditions) [22]. Since…”

Gevrey regularity for Navier–Stokes equations under Lions boundary conditions

“…In that case, we also recover (1) in the Euclidean case because the curvature vanishes. Writing the Navier-Stokes as (2), we are following [Ily91,Ily94,CRT99,FF05,Rod08,Rod07]; for other writings we refer to [Pri94,CF96].…”