Theodore D. Drivas

An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier–Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L3-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.

A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part II. Wall-bounded flows

We derive here Lagrangian fluctuation-dissipation relations for advected scalars in wall-bounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values, and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary local-time density at points on the wall where scalar fluxes are imposed and the boundary first hitting-time at points where scalar values are imposed. These concepts are illustrated both by analytical results for the problem of pure heat conduction and by numerical results from a database of channel-flow flow turbulence, which also demonstrate the scalar mixing properties of near-wall turbulence. As an application of the fluctuation-dissipation relation, we examine for wall-bounded flows the relation between anomalous scalar dissipation and Lagrangian spontaneous stochasticity, i.e. the persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. In the first paper of this series, we showed that spontaneous stochasticity is the only possible mechanism for anomalous dissipation of passive or active scalars, away from walls. Here it is shown that this remains true when there are no scalar fluxes through walls. Simple examples show, on the other hand, that a distinct mechanism of non-vanishing scalar dissipation can be thin scalar boundary layers near the walls. Nevertheless, we prove for general wall-bounded flows that spontaneous stochasticity is another possible mechanism of anomalous scalar dissipation.

Onsager's Conjecture and Anomalous Dissipation on Domains with Boundary

We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain

Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

\Omega\subset \mathbb{R}^d

Spontaneous Stochasticity and Anomalous Dissipation for Burgers Equation

,

Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind.

d\ge 2

Caustics and wave propagation in curved spacetimes

, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity

Dependence of self-force on central object

u\in L^3(0,T;B_{3}^{1/3, c_0})

A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls

. On an arbitrary thin neighborhood of the boundary, we assume boundedness of velocity and pressure and, at the boundary, we assume continuity of wall-normal velocity. We also prove two theorems which establish that the global viscous dissipation vanishes in the inviscid limit for Leray--Hopf solutions

Cascades and Dissipative Anomalies in Compressible Fluid Turbulence

u^\nu