We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.

1.
D.
Barbato
,
F.
Flandoli
, and
F.
Morandin
, “
A theorem of uniqueness for an inviscid dyadic model
,”
C. R. Math. Acad. Sci.
348
,
525
528
(
2010
).
2.
D.
Barbato
,
F.
Flandoli
, and
F.
Morandin
, “
Energy dissipation and self-similar solutions for an unforced inviscid dyadic model
,”
Trans. Am. Math. Soc.
363
,
1925
1946
(
2011
).
3.
D.
Barbato
,
F.
Morandin
, and
M.
Romito
, “
Smooth solutions for the dyadic model
,”
Nonlinearity
24
,
3083
(
2011
).
4.
A.
Cheskidov
and
S.
Friedlander
, “
The vanishing viscosity limit for a dyadic model
,”
J. Phys. D
238
,
783
787
(
2009
).
5.
A.
Cheskidov
,
S.
Friedlander
, and
N.
Pavlović
, “
Inviscid dyadic model of turbulence: the fixed point and Onsager's conjecture
,”
J. Math. Phys.
48
,
065503
(
2007
).
6.
A.
Cheskidov
,
S.
Friedlander
, and
N.
Pavlović
, “
An inviscid dyadic model of turbulence: The global attractor
,”
Discrete Contin. Dyn. Syst.
26
,
781
794
(
2010
).
7.
P.
Constantin
,
B.
Levant
, and
E. S.
Titi
, “
Analytic study of shell models of turbulence
,”
J. Phys. D
219
,
120
141
(
2006
).
8.
F.
Flandoli
,
Random perturbation of PDEs and fluid dynamic models
,
Lecture Notes in Mathematics
Vol.
2015
(
Springer
,
Heidelberg
,
2011
) pp.
x+176
.
9.
U.
Frisch
,
Turbulence
(
Cambridge University Press
,
Cambridge
,
1995
), pp.
xiv+296
.
10.
N. H.
Katz
and
N.
Pavlović
, “
Finite time blow-up for a dyadic model of the Euler equations
,”
Trans. Am. Math. Soc.
357
,
695
708
(
2005
).
11.
A.
Kiselev
and
A.
Zlatoš
, “
On discrete models of the Euler equation
,”
Int. Math. Res. Notices
38
,
2315
2339
(
2005
).
12.
A. N.
Kolmogorov
, “
The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers
,”
Dokl. Akad. Nauk SSSR
30
,
301
305
(
1941
)
A. N.
Kolmogorov
, [reprinted in
Proc. R. Soc. Lond. A
434
,
9
13
(
1991
)].
You do not currently have access to this content.