TY - JOUR

T1 - Onsager's conjecture and anomalous dissipation on domains with boundary

AU - Drivas, Theodore D.

AU - Nguyen, Huy Q.

N1 - Funding Information:
∗Received by the editors April 4, 2018; accepted for publication (in revised form) June 26, 2018; published electronically September 6, 2018. http://www.siam.org/journals/sima/50-5/M117886.html Funding: The first author was partially supported by NSF-DMS grant 1703997. †Department of Mathematics, Princeton University, Princeton, NJ 08544 (tdrivas@math. princeton.edu, qn@math.princeton.edu).
Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

PY - 2018

Y1 - 2018

N2 - We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain ω ⊂ ℝd, d ≥ 2, with boundary. In the bulk of uid, we assume Besov regularity of the velocity u ϵ L3(0, T;B1=3,c0 3 ). 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 uν of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a thin boundary layer of width O(ν min-1, 1/2(1-σ) }) when u ϵ L3(0, T;Bγ,c0 3 ) in the interior for any σ ϵ [1=3; 1]. The first theorem assumes continuity of the velocity in the boundary layer, whereas the second assumes a condition on the vanishing of energy dissipation within the layer. In both cases, strong L3t L3 x,loc convergence holds to a weak solution of the Euler equations. Finally, if a strong Euler solution exists in the background, we show that equicontinuity at the boundary within a O(v) strip alone suffices to conclude the absence of anomalous dissipation.

AB - We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain ω ⊂ ℝd, d ≥ 2, with boundary. In the bulk of uid, we assume Besov regularity of the velocity u ϵ L3(0, T;B1=3,c0 3 ). 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 uν of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a thin boundary layer of width O(ν min-1, 1/2(1-σ) }) when u ϵ L3(0, T;Bγ,c0 3 ) in the interior for any σ ϵ [1=3; 1]. The first theorem assumes continuity of the velocity in the boundary layer, whereas the second assumes a condition on the vanishing of energy dissipation within the layer. In both cases, strong L3t L3 x,loc convergence holds to a weak solution of the Euler equations. Finally, if a strong Euler solution exists in the background, we show that equicontinuity at the boundary within a O(v) strip alone suffices to conclude the absence of anomalous dissipation.

KW - Anomalous dissipation

KW - Bounded domain

KW - Inviscid limit

KW - Onsager's conjecture

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U2 - 10.1137/18M1178864

DO - 10.1137/18M1178864

M3 - Article

AN - SCOPUS:85055346732

SN - 0036-1410

VL - 50

SP - 4785

EP - 4811

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 5

ER -