Zoran Grujić

SPATIAL ANALYTICITY PROPERTIES OF NONLINEAR WAVES

In this paper, we study spatial analyticity properties of two classes of equations modeling unidirectional waves in nonlinear, dispersive media, namely KdV-type equations and BBM-type equations. The commentary begins with KdV-type equations and the observation that, for a class of such equations, boundedness of a solution suffices to maintain analyticity and so loss of analyticity detects loss of L∞-regularity. For a larger class of KdV-type equations, the same conclusion is valid provided that L∞-boundedness of a solution is replaced by -boundedness. It is also shown that these nonlinear dispersive wave equations are amenable to Gevrey-class analysis based on the boundedness of a Sobolev norm. This analysis yields an explicit lower bound on the possible rate of decrease in time of the uniform radius of analyticity of a solution in terms of the assumed Sobolev bound and the Gevrey-norm of the initial data. Attention is then shifted to BBM-type equations. It is shown that, regardless of the strength of the nonlinearity, a solution starting in a Gevrey space remains in this class for all time. Moreover, a lower bound on the possible rate of decrease in time of the uniform analyticity radius has temporal asymptotics that are independent of the degree of the nonlinearity, and so apparently determined in the main by the dispersion.

Coherent Vortex Structures and 3D Enstrophy Cascade

Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities –

A geometric measure-type regularity criterion for solutions to the 3D Navier–Stokes equations

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Energy Cascades and Flux Locality in Physical Scales of the 3D Navier-Stokes Equations

-Hölder coherence of the vorticity direction – coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.

A remark on time-analyticity for the Kuramoto-Sivashinsky equation

For the Kuramoto–Sivashinsky equation with L-periodic boundary conditions we show that the radius of space analyticity on the global attractor is lower-semicontinuous function at the stationary solutions, and thereby deduce the existence of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter L. As an application of the result, we prove that the number of rapid spatial oscillations of functions belonging to this neighborhood is, up to a logarithmic correction, at most linear in L.

Anomalous Dissipation and Energy Cascade in 3D Inviscid Flows

A local anisotropic geometric measure-type condition on the super-level sets of solutions to the 3D NSE preventing the formation of finite-time singularity is presented; essentially, local one-dimensional sparseness of the regions of intense fluid activity in a very weak sense.

Blow-Up Scenarios for the 3D Navier–Stokes Equations Exhibiting Sub-Criticality with Respect to the Scaling of One-Dimensional Local Sparseness

A mathematical evidence—in a statistically significant sense—of a geometric scenario leading to criticality of the Navier-Stokes problem is presented.

ON THE TRANSPORT AND CONCENTRATION OF ENSTROPHY IN 3D MAGNETOHYDRODYNAMIC TURBULENCE.

Rigorous estimates for the total – (kinetic) energy plus pressure – flux in