Dissipation in Onsager’s Critical Classes and Energy Conservation in BV∩L∞ with and Without Boundary

• Published in: Communications in mathematical physics Vol. 405; no. 1
• Main Authors: De Rosa, Luigi, Inversi, Marco
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Complex Systems; Compressibility; Conservation; Energy conservation; Euler-Lagrange equation; Fields (mathematics); Flavors; Fluid flow; Incompressibility; Incompressible flow; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Regularization; Relativity Theory; Singularity (mathematics); Theoretical; Vortex sheets
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-023-04922-3

This paper is concerned with the incompressible Euler equations. In Onsager’s critical classes we provide explicit formulas for the Duchon–Robert measure in terms of the regularization kernel and a family of vector-valued measures
{
μ
z
}
z
⊂
M
x
,
t
, having some Hölder regularity with respect to the direction
z
∈
B
1
. Then, we prove energy conservation for
L
x
,
t
∞
∩
L
t
1
B
V
x
solutions, in both the absence or presence of a physical boundary. This result generalises the previously known case of Vortex Sheets, showing that energy conservation follows from the structure of
L
∞
∩
B
V
incompressible vector fields rather than the flow having “organized singularities”. The interior energy conservation features the use of Ambrosio’s anisotropic optimization of the convolution kernel and it differs from the usual energy conservation arguments by heavily relying on the incompressibility of the vector field. This is the first energy conservation proof, for a given class of solutions, which fails to simultaneously apply to both compressible and incompressible models, coherently with compressible shocks having non-trivial entropy production. To run the boundary analysis we introduce a notion of “normal Lebesgue trace” for general vector fields, very reminiscent of the one for
BV
functions. We show that having such a null normal trace is basically equivalent to have vanishing boundary energy flux. This goes beyond the previous approaches, laying down a setup which applies to every Lipschitz bounded domain. Allowing any Lipschitz boundary introduces several technicalities to the proof, with a quite geometrical/measure-theoretical flavour.