Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations

By J. Duchon, R. Robert. 2000

426 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/fcfa1816d57768e64c5c7341aed26e71f439cbf1

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

The Equations of Magnetohydrodynamics: On the Interaction Between Matter and Radiation in the Evolution of Gaseous Stars (2006, background): “If the motion is not (known to be) smooth, however, the validity of (1.20) is no longer guaranteed, in particular, the dissipation rate of the mechanical energy may exceed the value S :∇ x u (see, for instance, [ 12 , 16] to illuminate this interesting and still largely open question).”
Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations (2016, neutral): “This is well-known (see for instance [12, 16]).”
On Nonperiodic Euler Flows with Hölder Regularity (2014, background): “The conservation of energy stated by Onsager was proven in [7,18], and this result was refined in [6] to show that energy conservation holds for energy class solutions in the space L(3)t B 1/3 3,c0(N) on either I × Tn or I × Rn (see also [16,21] for further proofs).”
Dissipative Euler Flows and Onsager's Conjecture (2012, background): “Slightly weaker assumptions on v (in Besov spaces) were subsequently shown to be sufficient for energy conservation in [16, 6].”
Dissipative Euler Flows with Onsager‐Critical Spatial Regularity (2014, background): “; t / 1=3C" WD sup t2Œ0;1 sup x¤y jv.x; t/ v.y; t/j jx yj 1=3C" <1; then the total kinetic energy is constant: E.t/ D E.0/.”
Lectures on the Onsager conjecture (2010, methodology): “The end result of a series of works by Constantin et al [8, 14], Eyink [21], and Duchon and Robert [ 20 ] was the proof of the Onsager conjecture in the Besov class B 1/3 3,c0 . In other words, a solution needs to be ”a little” smoother th en just 1/3.”
The $h$-principle and the equations of fluid dynamics (2011, background): “The latter has been rigorously proved with various sharper versions in [37, 24, 34], in particular we refer the reader to the recent survey [70].”
Weak Solutions with Decreasing Energy¶of Incompressible Euler Equations (2000, background): “I am also thankful to A. Kupiainen for an important reference [ DR ].”
Markov selections for the 3D stochastic Navier–Stokes equations (2006, background): “This difficulty cannot be overcome at the present state of underst anding of the NSE, see, for instance Constantin, E & Titi [5] and Duchon & Ro bert [10] on this and related technical problems.”
Onsager and the theory of hydrodynamic turbulence (2006, methodology): “The surprising answer seems to be that he had a valid proof in physical space and that it was virtually identical to that of Duchon and Robert (2000) fifty years later!”
On the Euler equations of incompressible fluids (2007, background): “One direction of the Onsager conjecture was addressed mathematically ([45], [73], [78]), and it was established that energy conservation occurs for u in Besov spaces B 3,∞, s > 1/3.”
Dissipation in Turbulent Flows (2015, methodology): “Nie & Tanveer (1999) (see also Duchon & Robert 2000, Eyink 2003) introduced the procedure of integrating both sides over a sphere of radius |r| = r , which, using the Gauss divergence theorem, yields∫ r̂· <δu|δu|2> d ≈ −16π 3 r, (5) where d is the differential of the solid angle in r space and r̂ ≡…”
Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows (2004, methodology): “As pointed out by Duchon and Robert [13, 63], it is remarkable that every solution of the Navier–Stokes equations obtained as a limit of (a subsequence of) solutions uε of the regularized equations introduced by Leray (3.6) is “dissipative” in the sense that D(u) ≥ 0; that is, these solutions of the”
Anomalous dissipation for 1/5-Hölder Euler flows (2015, background): “Slightly weaker assumptions on v (in Besov spaces) were subsequently shown to be sufficient for energy conservation in [12, 2].”