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In this paper we describe the long‐time behavior of the non‐cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (that is, infinite Knudsen number ). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator and its interplay with the singular collision operator. For ‐wavenumbers with , one sees anenhanced dissipationeffect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one seesTaylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay.

 

In this note we point out some simple sufficient (plausible) conditions for ‘turbulence’ cascades in suitable limits of damped, stochastically-driven nonlinear Schrödinger equation in ad-dimensional periodic box. Simple characterizations of dissipation anomalies for the wave action and kinetic energy in rough analogy with those that arise for fully developed turbulence in the 2D Navier–Stokes equations are given and sufficient conditions are given which differentiate between a ‘weak’ turbulence regime and a ‘strong’ turbulence regime. The proofs are relatively straightforward once the statements are identified, but we hope that it might be useful for thinking about mathematically precise formulations of the statistically-stationary wave turbulence problem. 

We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies$$N\ge 392$$$N\ge 392$. By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

 

We consider solutions of the Navier‐Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence‐free vector‐valued measure of arbitrary mass supported on a smooth curve. First, we prove global well‐posedness for perturbations of the Oseen vortex column in scaling‐critical spaces. Second, we prove local well‐posedness (in a sense to be made precise) when the filament is a smooth, closed, non‐self‐intersecting curve. Besides their physical interest, these results are the first to give well‐posedness in a neighborhood of large self‐similar solutions of 3d Navier‐Stokes, as well as solutions that are locally approximately self‐similar. © 2023 Wiley Periodicals LLC.

 

This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re . In this work, we show that there is constant 0 > c 0 ≪ 1 0 > c_0 \ll 1 , independent of R e \mathbf {Re} , such that sufficiently regular disturbances of size ϵ ≲ R e − 2 / 3 − δ \epsilon \lesssim \mathbf {Re}^{-2/3-\delta } for any δ > 0 \delta > 0 exist at least until t = c 0 ϵ − 1 t = c_0\epsilon ^{-1} and in general evolve to be O ( c 0 ) O(c_0) due to the lift-up effect. Further, after times t ≳ R e 1 / 3 t \gtrsim \mathbf {Re}^{1/3} , the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ϵ − 1 t \approx \epsilon ^{-1} . Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ \Rightarrow streak growth ⇒ \Rightarrow streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature. 

We investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.