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## Anomalous dissipation, anomalous work, and energy balance for the navier-stokes equations

Publication ,  Journal Article
CHESKIDOV, A; LUO, X
Published in: Siam Journal on Mathematical Analysis
January 1, 2021

In this paper, we study the energy balance for a class of solutions of the Navier-Stokes equations with external forces in dimensions three and above. The solution and force are smooth on (0, T) and the total dissipation and work of the force are both finite. We show that a possible failure of the energy balance stems from two effects. The first is the anomalous dissipation of the solution, which has been studied in many contexts. The second is what we call the anomalous work done by the force, a phenomenon that has not been analyzed before. There are numerous examples of solutions exhibiting anomalous work, for which even a continuous energy profile does not rule out the anomalous dissipation, but only implies the balance of the strengths of these two effects, which we confirm in explicit constructions. More importantly, we show that there exist solutions exhibiting anomalous dissipation with zero anomalous work. Hence the violation of the energy balance results from the nonlinearity of the solution instead of artifacts of the force. Such examples exist in the class u ∈ L3t B 1 3 3,7infty; and f ∈ L2 t H 1, which implies the sharpness of many existing conditions on the energy balance.

## Published In

Siam Journal on Mathematical Analysis

1095-7154

0036-1410

January 1, 2021

53

4

## Start / End Page

3856 / 3887

• Applied Mathematics
• 0102 Applied Mathematics
• 0101 Pure Mathematics

### Citation

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CHESKIDOV, A., & LUO, X. (2021). Anomalous dissipation, anomalous work, and energy balance for the navier-stokes equations. Siam Journal on Mathematical Analysis, 53(4), 3856–3887. https://doi.org/10.1137/20M1354076
CHESKIDOV, A., and X. LUO. “Anomalous dissipation, anomalous work, and energy balance for the navier-stokes equations.” Siam Journal on Mathematical Analysis 53, no. 4 (January 1, 2021): 3856–87. https://doi.org/10.1137/20M1354076.
CHESKIDOV A, LUO X. Anomalous dissipation, anomalous work, and energy balance for the navier-stokes equations. Siam Journal on Mathematical Analysis. 2021 Jan 1;53(4):3856–87.
CHESKIDOV, A., and X. LUO. “Anomalous dissipation, anomalous work, and energy balance for the navier-stokes equations.” Siam Journal on Mathematical Analysis, vol. 53, no. 4, Jan. 2021, pp. 3856–87. Scopus, doi:10.1137/20M1354076.
CHESKIDOV A, LUO X. Anomalous dissipation, anomalous work, and energy balance for the navier-stokes equations. Siam Journal on Mathematical Analysis. 2021 Jan 1;53(4):3856–3887.

## Published In

Siam Journal on Mathematical Analysis

1095-7154

0036-1410

January 1, 2021

53

4

3856 / 3887